| 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-c4 |
72.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$1.37737$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$9.301119475$ |
0.878873180 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 293 a - 770\) , \( 4024 a - 10642\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(293a-770\right){x}+4024a-10642$ |
| 72.1-d4 |
72.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$1.37737$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$22.73403407$ |
2.148164301 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 292 a - 771\) , \( -4503 a + 11915\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(292a-771\right){x}-4503a+11915$ |
| 432.1-c4 |
432.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{14} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$8.395474317$ |
1.586595513 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 23 a - 70\) , \( -86 a + 223\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(23a-70\right){x}-86a+223$ |
| 432.1-g4 |
432.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{14} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.643081635$ |
$8.395474317$ |
4.081241752 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 25 a - 65\) , \( 110 a - 291\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(25a-65\right){x}+110a-291$ |
| 432.2-c4 |
432.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.2 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{14} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$8.395474317$ |
1.586595513 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -25 a - 70\) , \( 85 a + 223\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-25a-70\right){x}+85a+223$ |
| 432.2-g4 |
432.2-g |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.2 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{14} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.643081635$ |
$8.395474317$ |
4.081241752 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -25 a - 65\) , \( -110 a - 291\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-25a-65\right){x}-110a-291$ |
| 648.1-f4 |
648.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{20} \) |
$2.38568$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.818877155$ |
$3.100373158$ |
3.838342240 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 2630 a - 6949\) , \( 112947 a - 298821\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2630a-6949\right){x}+112947a-298821$ |
| 648.1-m4 |
648.1-m |
$6$ |
$8$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{20} \) |
$2.38568$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.322644888$ |
$7.578011356$ |
3.696502570 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 2631 a - 6945\) , \( -114640 a + 303321\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2631a-6945\right){x}-114640a+303321$ |
*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.
