| 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 |
| 20.3-a3 |
20.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{19} \cdot 5 \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$2.778323884$ |
$7.344894602$ |
0.814469976 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -4 a - 4\) , \( 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-4a-4\right){x}+16$ |
| 100.4-a3 |
100.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
100.4 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{7} \cdot 5^{7} \) |
$1.57333$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.284736723$ |
2.359824525 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -a - 8\) , \( -a + 8\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(-a-8\right){x}-a+8$ |
| 500.7-d3 |
500.7-d |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
500.7 |
\( 2^{2} \cdot 5^{3} \) |
\( 2^{19} \cdot 5^{7} \) |
$2.35267$ |
$(2,a), (2,a+1), (5,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.539796968$ |
$3.284736723$ |
3.821478377 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -7 a + 24\) , \( 15 a + 20\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(-7a+24\right){x}+15a+20$ |
| 640.13-c3 |
640.13-c |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.13 |
\( 2^{7} \cdot 5 \) |
\( 2^{37} \cdot 5 \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.596812390$ |
1.865605094 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 13 a - 26\) , \( 31 a - 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a+1\right){x}^2+\left(13a-26\right){x}+31a-4$ |
| 640.3-c3 |
640.3-c |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.3 |
\( 2^{7} \cdot 5 \) |
\( 2^{37} \cdot 5 \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1.599780362$ |
$2.596812390$ |
2.984558394 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 14 a - 1\) , \( -17 a - 33\bigr] \) |
${y}^2={x}^3+\left(-a+1\right){x}^2+\left(14a-1\right){x}-17a-33$ |
| 1280.9-d3 |
1280.9-d |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1280.9 |
\( 2^{8} \cdot 5 \) |
\( 2^{43} \cdot 5 \) |
$2.97592$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.654672367$ |
$1.836223650$ |
4.365628051 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -28 a + 24\) , \( 56 a - 336\bigr] \) |
${y}^2={x}^3+\left(-a-1\right){x}^2+\left(-28a+24\right){x}+56a-336$ |
| 1620.3-b3 |
1620.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1620.3 |
\( 2^{2} \cdot 3^{4} \cdot 5 \) |
\( 2^{19} \cdot 3^{12} \cdot 5 \) |
$3.15644$ |
$(2,a), (2,a+1), (5,a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$2.448298200$ |
5.276728053 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -18 a + 17\) , \( 22 a - 75\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+{x}^2+\left(-18a+17\right){x}+22a-75$ |
| 2500.8-j3 |
2500.8-j |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
2500.8 |
\( 2^{2} \cdot 5^{4} \) |
\( 2^{19} \cdot 5^{13} \) |
$3.51807$ |
$(2,a), (2,a+1), (5,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.994731198$ |
$1.468978920$ |
12.63078489 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -45 a + 42\) , \( -62 a + 436\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3-a{x}^2+\left(-45a+42\right){x}-62a+436$ |
| 3200.19-c3 |
3200.19-c |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.19 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{25} \cdot 5^{7} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.213747773$ |
$1.161329805$ |
2.025317702 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 10 a + 35\) , \( 22 a - 168\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(10a+35\right){x}+22a-168$ |
| 3200.4-c3 |
3200.4-c |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.4 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{25} \cdot 5^{7} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.161329805$ |
1.668647924 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 52\) , \( -42 a\bigr] \) |
${y}^2+a{x}{y}={x}^3+52{x}-42a$ |
*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.
