| 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 |
| 10.1-a3 |
10.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{2} \) |
$0.88475$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2 \) |
$1$ |
$9.699765812$ |
1.742129368 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( a - 3\) , \( 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(a-3\right){x}+1$ |
| 50.1-a3 |
50.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{13} \cdot 5^{8} \) |
$1.32301$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.337867144$ |
1.558207877 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( a + 8\) , \( -8 a\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(a+8\right){x}-8a$ |
| 250.3-a3 |
250.3-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
250.3 |
\( 2 \cdot 5^{3} \) |
\( 2^{13} \cdot 5^{8} \) |
$1.97836$ |
$(2,a), (5,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.337867144$ |
1.558207877 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -a - 15\) , \( -3 a - 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-a-15\right){x}-3a-5$ |
| 320.1-b3 |
320.1-b |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
320.1 |
\( 2^{6} \cdot 5 \) |
\( 2^{19} \cdot 5^{2} \) |
$2.10430$ |
$(2,a), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{3} \) |
$0.655495815$ |
$3.429385091$ |
3.229946426 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -2 a - 1\) , \( 7\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-2a-1\right){x}+7$ |
| 640.13-h3 |
640.13-h |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.13 |
\( 2^{7} \cdot 5 \) |
\( 2^{19} \cdot 5^{2} \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{3} \) |
$1.125988460$ |
$3.429385091$ |
5.548292329 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -4 a + 6\) , \( 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-4a+6\right){x}+7$ |
| 810.1-a3 |
810.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
810.1 |
\( 2 \cdot 3^{4} \cdot 5 \) |
\( 2 \cdot 3^{12} \cdot 5^{2} \) |
$2.65424$ |
$(2,a), (5,a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.233255270$ |
1.161419578 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( a\) , \( 6\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3-{x}^2+a{x}+6$ |
| 1250.3-a3 |
1250.3-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1250.3 |
\( 2 \cdot 5^{4} \) |
\( 2 \cdot 5^{14} \) |
$2.95833$ |
$(2,a), (5,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{3} \) |
$0.317825018$ |
$1.939953162$ |
0.885907678 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 4 a - 3\) , \( -3 a - 29\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(4a-3\right){x}-3a-29$ |
| 1280.9-c3 |
1280.9-c |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1280.9 |
\( 2^{8} \cdot 5 \) |
\( 2^{25} \cdot 5^{2} \) |
$2.97592$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{5} \) |
$0.179307837$ |
$2.424941453$ |
2.499019599 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 3 a - 3\) , \( 2 a + 2\bigr] \) |
${y}^2={x}^3+a{x}^2+\left(3a-3\right){x}+2a+2$ |
| 1600.1-a3 |
1600.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1600.1 |
\( 2^{6} \cdot 5^{2} \) |
\( 2^{31} \cdot 5^{8} \) |
$3.14666$ |
$(2,a), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{4} \) |
$0.425631153$ |
$1.533667636$ |
1.875874574 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 5 a - 77\) , \( 34 a - 267\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a-1\right){x}^2+\left(5a-77\right){x}+34a-267$ |
| 3200.19-h3 |
3200.19-h |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.19 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{31} \cdot 5^{8} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.533667636$ |
2.203638713 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -12 a - 64\) , \( 36 a + 332\bigr] \) |
${y}^2={x}^3+\left(a-1\right){x}^2+\left(-12a-64\right){x}+36a+332$ |
| 3920.1-b3 |
3920.1-b |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3920.1 |
\( 2^{4} \cdot 5 \cdot 7^{2} \) |
\( 2^{25} \cdot 5^{2} \cdot 7^{6} \) |
$3.93678$ |
$(2,a), (5,a+1), (7,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{3} \) |
$0.599539385$ |
$1.833083436$ |
1.579098029 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -8 a + 59\) , \( 63 a + 69\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-8a+59\right){x}+63a+69$ |
| 3920.3-a3 |
3920.3-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3920.3 |
\( 2^{4} \cdot 5 \cdot 7^{2} \) |
\( 2^{13} \cdot 5^{2} \cdot 7^{6} \) |
$3.93678$ |
$(2,a), (5,a+1), (7,a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.833083436$ |
1.316926017 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 2 a - 8\) , \( -10 a + 20\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(2a-8\right){x}-10a+20$ |
| 4050.1-c3 |
4050.1-c |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
4050.1 |
\( 2 \cdot 3^{4} \cdot 5^{2} \) |
\( 2^{13} \cdot 3^{12} \cdot 5^{8} \) |
$3.96902$ |
$(2,a), (5,a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{3} \) |
$2.539773523$ |
$1.445955714$ |
5.276660147 |
\( -\frac{19951}{50} a - \frac{47943}{50} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 4 a + 88\) , \( 123 a - 124\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(4a+88\right){x}+123a-124$ |
*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.
