{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T15:54:03Z","timestamp":1776873243739,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"313","license":[{"start":{"date-parts":[[2018,11,16]],"date-time":"2018-11-16T00:00:00Z","timestamp":1542326400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Assuming Generalized Riemann\u2019s Hypothesis, Bach proved that the class group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper C script l Subscript bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:msub>\n                              <mml:mi>\n                                \u2113\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal C\\!\\ell _{\\mathbf {K}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of a number field\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {K}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    may be generated using prime ideals whose norm is bounded by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"12 log squared normal upper Delta Subscript bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>12<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">12\\log ^2\\Delta _{\\mathbf {K}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 4 plus o left-parenthesis 1 right-parenthesis right-parenthesis log squared normal upper Delta Subscript bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>o<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(4+o(1))\\log ^2\\Delta _{\\mathbf {K}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    asymptotically, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta Subscript bold upper K\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Delta _{\\mathbf {K}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the absolute value of the discriminant of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {K}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper C script l Subscript bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:msub>\n                              <mml:mi>\n                                \u2113\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal C\\!\\ell _{\\mathbf {K}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and which performs better than Bach\u2019s bound in computations, but which is asymptotically worse. In this paper we show that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper C script l Subscript bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:msub>\n                              <mml:mi>\n                                \u2113\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal C\\!\\ell _{\\mathbf {K}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is generated by prime ideals whose norm is bounded by the minimum of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4.01 log squared normal upper Delta Subscript bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4.01<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4.01\\log ^2\\Delta _{\\mathbf {K}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 left-parenthesis 1 plus left-parenthesis 2 pi e Superscript gamma Baseline right-parenthesis Superscript minus upper N Super Subscript bold upper K Superscript Baseline right-parenthesis squared log squared normal upper Delta Subscript bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mstyle scriptlevel=\"0\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mstyle>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mstyle scriptlevel=\"0\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mstyle>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>e<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03b3\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>N<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                    <\/mml:mrow>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mstyle scriptlevel=\"0\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)<\/mml:mo>\n                                <\/mml:mrow>\n                              <\/mml:mstyle>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4\\big (1+\\big (2\\pi e^{\\gamma })^{-N_{\\mathbf {K}}}\\big )^2\\log ^2\\Delta _{\\mathbf {K}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 left-parenthesis log normal upper Delta Subscript bold upper K Baseline plus log log normal upper Delta Subscript bold upper K Baseline minus left-parenthesis gamma plus log 2 pi right-parenthesis upper N Subscript bold upper K Baseline plus 1 plus left-parenthesis upper N Subscript bold upper K Baseline plus 1 right-parenthesis StartFraction log left-parenthesis 7 log normal upper Delta Subscript bold upper K Baseline right-parenthesis Over log normal upper Delta Subscript bold upper K Baseline EndFraction right-parenthesis squared\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mstyle scriptlevel=\"0\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mstyle>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b3\n                              \n                            <\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>N<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>N<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>log<\/mml:mi>\n                                <mml:mo>\n                                  \u2061\n                                  \n                                <\/mml:mo>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mn>7<\/mml:mn>\n                                <mml:mi>log<\/mml:mi>\n                                <mml:mo>\n                                  \u2061\n                                  \n                                <\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi mathvariant=\"normal\">\n                                    \u0394\n                                    \n                                  <\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                    <\/mml:mrow>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mi>log<\/mml:mi>\n                                <mml:mo>\n                                  \u2061\n                                  \n                                <\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi mathvariant=\"normal\">\n                                    \u0394\n                                    \n                                  <\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                    <\/mml:mrow>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                            <mml:msup>\n                              <mml:mstyle scriptlevel=\"0\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)<\/mml:mo>\n                                <\/mml:mrow>\n                              <\/mml:mstyle>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4\\big (\\log \\Delta _{\\mathbf {K}} +\\log \\log \\Delta _{\\mathbf {K}}-(\\gamma +\\log 2\\pi )N_{\\mathbf {K}}+1+(N_{\\mathbf {K}}+1)\\frac {\\log (7\\log \\Delta _{\\mathbf {K}})} {\\log \\Delta _{\\mathbf {K}}}\\big )^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman\u2019s algorithms, confirming that it has size\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"equivalent-to left-parenthesis log normal upper Delta Subscript bold upper K Baseline log log normal upper Delta Subscript bold upper K Baseline right-parenthesis squared\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u224d\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\asymp (\\log \\Delta _{\\mathbf {K}}\\log \\log \\Delta _{\\mathbf {K}})^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"log squared normal upper Delta Subscript bold upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u0394\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\log ^2\\Delta _{\\mathbf {K}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    except for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"7\">\n                        <mml:semantics>\n                          <mml:mn>7<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">7<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    out of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"31292\">\n                        <mml:semantics>\n                          <mml:mn>31292<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">31292<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    fields we tested.\n                  <\/p>","DOI":"10.1090\/mcom\/3281","type":"journal-article","created":{"date-parts":[[2017,4,20]],"date-time":"2017-04-20T09:41:19Z","timestamp":1492681279000},"page":"2483-2511","source":"Crossref","is-referenced-by-count":4,"title":["Explicit bounds for generators of the class group"],"prefix":"10.1090","volume":"87","author":[{"given":"Lo\u00efc","family":"Greni\u00e9","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giuseppe","family":"Molteni","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2017,11,16]]},"reference":[{"issue":"191","key":"1","doi-asserted-by":"publisher","first-page":"355","DOI":"10.2307\/2008811","article-title":"Explicit bounds for primality testing and related problems","volume":"55","author":"Bach, Eric","year":"1990","journal-title":"Math. 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