| 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 |
| 5.1-a1 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( -5 \) |
$2.48194$ |
$(14a+123)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$22.47775106$ |
1.210161437 |
\( -\frac{32886}{5} a + \frac{329913}{5} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 1554151062 a - 15210612881\) , \( 100918511605110 a - 987698334784476\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(1554151062a-15210612881\right){x}+100918511605110a-987698334784476$ |
| 5.1-a2 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{2} \) |
$2.48194$ |
$(14a+123)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$44.95550212$ |
1.210161437 |
\( \frac{9261}{5} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 20554656716310 a - 201170230197103\) , \( -40029828255758796894 a + 391775444178479697450\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(20554656716310a-201170230197103\right){x}-40029828255758796894a+391775444178479697450$ |
| 5.1-a3 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( -5 \) |
$2.48194$ |
$(14a+123)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$22.47775106$ |
1.210161437 |
\( \frac{32886}{5} a + \frac{297027}{5} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -4943291993102 a + 48380432809481\) , \( -4914040966528072718 a + 48094150443821025578\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-4943291993102a+48380432809481\right){x}-4914040966528072718a+48094150443821025578$ |
| 5.1-a4 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{4} \) |
$2.48194$ |
$(14a+123)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$44.95550212$ |
1.210161437 |
\( \frac{17779581}{25} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 255465019188385 a - 2500258575308473\) , \( -6765060610145139924639 a + 66210242234867888272904\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(255465019188385a-2500258575308473\right){x}-6765060610145139924639a+66210242234867888272904$ |
| 5.1-b1 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 2^{12} \cdot 5 \) |
$2.48194$ |
$(14a+123)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.195455604$ |
$39.10504687$ |
2.311095686 |
\( -\frac{32886}{5} a + \frac{329913}{5} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 224087337489 a + 1969075111736\) , \( -47426220973806953 a - 416738368217177607\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(224087337489a+1969075111736\right){x}-47426220973806953a-416738368217177607$ |
| 5.1-b2 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$2.48194$ |
$(14a+123)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$4.390911208$ |
$39.10504687$ |
2.311095686 |
\( \frac{9261}{5} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -58235967836 a - 511724563071\) , \( -6036632939936280 a - 53044423722997080\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-58235967836a-511724563071\right){x}-6036632939936280a-53044423722997080$ |
| 5.1-b3 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 2^{12} \cdot 5 \) |
$2.48194$ |
$(14a+123)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$8.781822416$ |
$39.10504687$ |
2.311095686 |
\( \frac{32886}{5} a + \frac{297027}{5} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -70452137 a - 619069116\) , \( 973280661195 a + 8552302634209\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-70452137a-619069116\right){x}+973280661195a+8552302634209$ |
| 5.1-b4 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{4} \) |
$2.48194$ |
$(14a+123)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.195455604$ |
$9.776261717$ |
2.311095686 |
\( \frac{17779581}{25} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -723789885956 a - 6360005284216\) , \( -1020216914157134148 a - 8964735610459501129\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-723789885956a-6360005284216\right){x}-1020216914157134148a-8964735610459501129$ |
| 5.1-c1 |
5.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( -5 \) |
$2.48194$ |
$(14a+123)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$22.47775106$ |
1.210161437 |
\( -\frac{32886}{5} a + \frac{329913}{5} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 8 a - 54\) , \( 66 a - 669\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(8a-54\right){x}+66a-669$ |
| 5.1-c2 |
5.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{2} \) |
$2.48194$ |
$(14a+123)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$44.95550212$ |
1.210161437 |
\( \frac{9261}{5} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 112434 a - 1100070\) , \( -15796441 a + 154602553\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(112434a-1100070\right){x}-15796441a+154602553$ |
| 5.1-c3 |
5.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( -5 \) |
$2.48194$ |
$(14a+123)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$22.47775106$ |
1.210161437 |
\( \frac{32886}{5} a + \frac{297027}{5} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -27036 a + 264628\) , \( -2078002 a + 20337565\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-27036a+264628\right){x}-2078002a+20337565$ |
| 5.1-c4 |
5.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{4} \) |
$2.48194$ |
$(14a+123)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$44.95550212$ |
1.210161437 |
\( \frac{17779581}{25} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 1397189 a - 13674080\) , \( -2731309655 a + 26731568829\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(1397189a-13674080\right){x}-2731309655a+26731568829$ |
| 5.1-d1 |
5.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 2^{12} \cdot 5 \) |
$2.48194$ |
$(14a+123)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$8.781822416$ |
$39.10504687$ |
2.311095686 |
\( -\frac{32886}{5} a + \frac{329913}{5} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 1229 a + 10911\) , \( -9797 a - 85860\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(1229a+10911\right){x}-9797a-85860$ |
| 5.1-d2 |
5.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$2.48194$ |
$(14a+123)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$4.390911208$ |
$39.10504687$ |
2.311095686 |
\( \frac{9261}{5} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -318 a - 2770\) , \( -3504 a - 30813\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-318a-2770\right){x}-3504a-30813$ |
| 5.1-d3 |
5.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 2^{12} \cdot 5 \) |
$2.48194$ |
$(14a+123)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.195455604$ |
$39.10504687$ |
2.311095686 |
\( \frac{32886}{5} a + \frac{297027}{5} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 3 a + 139\) , \( 9 a + 312\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(3a+139\right){x}+9a+312$ |
| 5.1-d4 |
5.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{345}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{4} \) |
$2.48194$ |
$(14a+123)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.195455604$ |
$9.776261717$ |
2.311095686 |
\( \frac{17779581}{25} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -3958 a - 34755\) , \( -425880 a - 3742268\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-3958a-34755\right){x}-425880a-3742268$ |
*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.
