| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 2.1-a1 |
2.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2 \cdot 3^{12} \) |
$2.35710$ |
$(-a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$24.00491108$ |
1.082224970 |
\( -\frac{14318673286331251}{2} a - \frac{158801768802767239}{2} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( 24784 a - 274563\) , \( -6814515 a + 75577659\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(24784a-274563\right){x}-6814515a+75577659$ |
| 2.1-a2 |
2.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \cdot 3^{12} \) |
$2.35710$ |
$(-a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$24.00491108$ |
1.082224970 |
\( -\frac{437921}{8} a + \frac{4840139}{8} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( 979 a - 10548\) , \( 65571 a - 726141\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(979a-10548\right){x}+65571a-726141$ |
| 2.1-b1 |
2.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( -2 \) |
$2.35710$ |
$(-a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$24.00491108$ |
1.082224970 |
\( -\frac{14318673286331251}{2} a - \frac{158801768802767239}{2} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 2754 a - 30501\) , \( -271745 a + 3013850\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(2754a-30501\right){x}-271745a+3013850$ |
| 2.1-b2 |
2.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \) |
$2.35710$ |
$(-a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$24.00491108$ |
1.082224970 |
\( -\frac{437921}{8} a + \frac{4840139}{8} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 109 a - 1166\) , \( 1668 a - 18445\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(109a-1166\right){x}+1668a-18445$ |
| 2.1-c1 |
2.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \cdot 3^{12} \) |
$2.35710$ |
$(-a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 5 \) |
$0.633258338$ |
$21.22402070$ |
6.059349823 |
\( \frac{437921}{8} a + \frac{4840139}{8} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -980 a - 10863\) , \( 55776 a + 618555\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-980a-10863\right){x}+55776a+618555$ |
| 2.1-c2 |
2.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2 \cdot 3^{12} \) |
$2.35710$ |
$(-a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$25$ |
\( 1 \) |
$3.166291692$ |
$0.848960828$ |
6.059349823 |
\( \frac{14318673286331251}{2} a - \frac{158801768802767239}{2} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -24785 a - 274878\) , \( -7062360 a - 78325395\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-24785a-274878\right){x}-7062360a-78325395$ |
| 2.1-d1 |
2.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \cdot 3^{12} \) |
$2.35710$ |
$(-a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$24.00491108$ |
1.082224970 |
\( \frac{437921}{8} a + \frac{4840139}{8} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -980 a - 10548\) , \( -65571 a - 726141\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-980a-10548\right){x}-65571a-726141$ |
| 2.1-d2 |
2.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2 \cdot 3^{12} \) |
$2.35710$ |
$(-a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$24.00491108$ |
1.082224970 |
\( \frac{14318673286331251}{2} a - \frac{158801768802767239}{2} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -24785 a - 274563\) , \( 6814515 a + 75577659\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-24785a-274563\right){x}+6814515a+75577659$ |
| 2.1-e1 |
2.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \) |
$2.35710$ |
$(-a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$24.00491108$ |
1.082224970 |
\( \frac{437921}{8} a + \frac{4840139}{8} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -109 a - 1166\) , \( -1668 a - 18445\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(-109a-1166\right){x}-1668a-18445$ |
| 2.1-e2 |
2.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( -2 \) |
$2.35710$ |
$(-a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$24.00491108$ |
1.082224970 |
\( \frac{14318673286331251}{2} a - \frac{158801768802767239}{2} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -2754 a - 30501\) , \( 271745 a + 3013850\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(-2754a-30501\right){x}+271745a+3013850$ |
| 2.1-f1 |
2.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( -2 \) |
$2.35710$ |
$(-a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$61.45608557$ |
$0.848960828$ |
4.704353957 |
\( -\frac{14318673286331251}{2} a - \frac{158801768802767239}{2} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 2734 a - 30206\) , \( 299185 a - 3317885\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2734a-30206\right){x}+299185a-3317885$ |
| 2.1-f2 |
2.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \) |
$2.35710$ |
$(-a-11)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$12.29121711$ |
$21.22402070$ |
4.704353957 |
\( -\frac{437921}{8} a + \frac{4840139}{8} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 89 a - 871\) , \( -678 a + 7760\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(89a-871\right){x}-678a+7760$ |
| 2.1-g1 |
2.1-g |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \) |
$2.35710$ |
$(-a-11)$ |
$0 \le r \le 1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
|
\( 5 \) |
$1$ |
$21.22402070$ |
4.704353957 |
\( \frac{437921}{8} a + \frac{4840139}{8} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -89 a - 871\) , \( 678 a + 7760\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-89a-871\right){x}+678a+7760$ |
| 2.1-g2 |
2.1-g |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( -2 \) |
$2.35710$ |
$(-a-11)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
|
\( 1 \) |
$1$ |
$0.848960828$ |
4.704353957 |
\( \frac{14318673286331251}{2} a - \frac{158801768802767239}{2} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -2734 a - 30206\) , \( -299185 a - 3317885\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2734a-30206\right){x}-299185a-3317885$ |
| 2.1-h1 |
2.1-h |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2 \cdot 3^{12} \) |
$2.35710$ |
$(-a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$25$ |
\( 1 \) |
$3.166291692$ |
$0.848960828$ |
6.059349823 |
\( -\frac{14318673286331251}{2} a - \frac{158801768802767239}{2} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 24784 a - 274878\) , \( 7062360 a - 78325395\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(24784a-274878\right){x}+7062360a-78325395$ |
| 2.1-h2 |
2.1-h |
$2$ |
$5$ |
\(\Q(\sqrt{123}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \cdot 3^{12} \) |
$2.35710$ |
$(-a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 5 \) |
$0.633258338$ |
$21.22402070$ |
6.059349823 |
\( -\frac{437921}{8} a + \frac{4840139}{8} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 979 a - 10863\) , \( -55776 a + 618555\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(979a-10863\right){x}-55776a+618555$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
