| 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-a4 |
20.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{33} \cdot 5^{3} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$0.926107961$ |
$2.448298200$ |
0.814469976 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 11 a + 21\) , \( -5 a + 45\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(11a+21\right){x}-5a+45$ |
| 100.4-a4 |
100.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
100.4 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{21} \cdot 5^{9} \) |
$1.57333$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$1.094912241$ |
2.359824525 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -11 a + 57\) , \( 63 a + 117\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(-11a+57\right){x}+63a+117$ |
| 500.7-d4 |
500.7-d |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
500.7 |
\( 2^{2} \cdot 5^{3} \) |
\( 2^{33} \cdot 5^{9} \) |
$2.35267$ |
$(2,a), (2,a+1), (5,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{3} \) |
$0.179932322$ |
$1.094912241$ |
3.821478377 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 88 a - 16\) , \( 204 a + 972\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(88a-16\right){x}+204a+972$ |
| 640.13-c4 |
640.13-c |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.13 |
\( 2^{7} \cdot 5 \) |
\( 2^{51} \cdot 5^{3} \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.865604130$ |
1.865605094 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -132 a - 81\) , \( 1054 a - 1099\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a+1\right){x}^2+\left(-132a-81\right){x}+1054a-1099$ |
| 640.3-c4 |
640.3-c |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.3 |
\( 2^{7} \cdot 5 \) |
\( 2^{51} \cdot 5^{3} \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$4.799341087$ |
$0.865604130$ |
2.984558394 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -66 a - 321\) , \( -545 a - 2145\bigr] \) |
${y}^2={x}^3+\left(-a+1\right){x}^2+\left(-66a-321\right){x}-545a-2145$ |
| 1280.9-d4 |
1280.9-d |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1280.9 |
\( 2^{8} \cdot 5 \) |
\( 2^{57} \cdot 5^{3} \) |
$2.97592$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$4.964017103$ |
$0.612074550$ |
4.365628051 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 212 a + 424\) , \( 696 a - 7632\bigr] \) |
${y}^2={x}^3+\left(-a-1\right){x}^2+\left(212a+424\right){x}+696a-7632$ |
| 1620.3-b4 |
1620.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1620.3 |
\( 2^{2} \cdot 3^{4} \cdot 5 \) |
\( 2^{33} \cdot 3^{12} \cdot 5^{3} \) |
$3.15644$ |
$(2,a), (2,a+1), (5,a+1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{4} \) |
$1$ |
$0.816099400$ |
5.276728053 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 117 a + 242\) , \( 472 a - 3693\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+{x}^2+\left(117a+242\right){x}+472a-3693$ |
| 2500.8-j4 |
2500.8-j |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
2500.8 |
\( 2^{2} \cdot 5^{4} \) |
\( 2^{33} \cdot 5^{15} \) |
$3.51807$ |
$(2,a), (2,a+1), (5,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{3} \) |
$0.664910399$ |
$0.489659640$ |
12.63078489 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 330 a + 667\) , \( -2187 a + 15561\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3-a{x}^2+\left(330a+667\right){x}-2187a+15561$ |
| 3200.19-c4 |
3200.19-c |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.19 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{39} \cdot 5^{9} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$3.641243320$ |
$0.387109935$ |
2.025317702 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 25 a - 500\) , \( 228 a - 3782\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(25a-500\right){x}+228a-3782$ |
| 3200.4-c4 |
3200.4-c |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.4 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{39} \cdot 5^{9} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$0.387109935$ |
1.668647924 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 135 a - 388\) , \( -1495 a + 1832\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(135a-388\right){x}-1495a+1832$ |
*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.
