| 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 |
| 72.1-c5 |
72.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$1.37737$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$37.20447790$ |
0.878873180 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 773 a - 2040\) , \( -19496 a + 51586\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(773a-2040\right){x}-19496a+51586$ |
| 72.1-d5 |
72.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$1.37737$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$5.683508517$ |
2.148164301 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 772 a - 2041\) , \( 18227 a - 48223\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(772a-2041\right){x}+18227a-48223$ |
| 432.1-c5 |
432.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{8} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$8.395474317$ |
1.586595513 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 63 a - 180\) , \( 522 a - 1377\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(63a-180\right){x}+522a-1377$ |
| 432.1-g5 |
432.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{8} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.643081635$ |
$8.395474317$ |
4.081241752 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 65 a - 175\) , \( -458 a + 1199\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(65a-175\right){x}-458a+1199$ |
| 432.2-c5 |
432.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.2 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{8} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$8.395474317$ |
1.586595513 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -65 a - 180\) , \( -523 a - 1377\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-65a-180\right){x}-523a-1377$ |
| 432.2-g5 |
432.2-g |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.2 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{8} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.643081635$ |
$8.395474317$ |
4.081241752 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -65 a - 175\) , \( 458 a + 1199\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-65a-175\right){x}+458a+1199$ |
| 648.1-f5 |
648.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{14} \) |
$2.38568$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.204719288$ |
$12.40149263$ |
3.838342240 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -6945 a - 18375\) , \( 496604 a + 1313891\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-6945a-18375\right){x}+496604a+1313891$ |
| 648.1-m5 |
648.1-m |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{14} \) |
$2.38568$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.290579554$ |
$1.894502839$ |
3.696502570 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -6946 a - 18379\) , \( -528879 a - 1399283\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-6946a-18379\right){x}-528879a-1399283$ |
*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.
