| 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-a1 |
10.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{26} \cdot 5^{7} \) |
$0.88475$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$1.385680830$ |
1.742129368 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -67 a + 360\) , \( 905 a + 1396\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2+\left(-67a+360\right){x}+905a+1396$ |
| 50.1-a1 |
50.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
50.1 |
\( 2 \cdot 5^{2} \) |
\( 2^{26} \cdot 5^{13} \) |
$1.32301$ |
$(2,a), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$0.619695306$ |
1.558207877 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -609 a + 1079\) , \( -674 a + 33097\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-609a+1079\right){x}-674a+33097$ |
| 250.3-a1 |
250.3-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
250.3 |
\( 2 \cdot 5^{3} \) |
\( 2^{14} \cdot 5^{13} \) |
$1.97836$ |
$(2,a), (5,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$0.619695306$ |
1.558207877 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 124 a + 245\) , \( -292 a + 3566\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(124a+245\right){x}-292a+3566$ |
| 320.1-b1 |
320.1-b |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
320.1 |
\( 2^{6} \cdot 5 \) |
\( 2^{44} \cdot 5^{7} \) |
$2.10430$ |
$(2,a), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$9.176941414$ |
$0.489912155$ |
3.229946426 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 808 a - 2341\) , \( -22422 a + 26059\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(808a-2341\right){x}-22422a+26059$ |
| 640.13-h1 |
640.13-h |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.13 |
\( 2^{7} \cdot 5 \) |
\( 2^{44} \cdot 5^{7} \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \cdot 7^{2} \) |
$0.321710988$ |
$0.489912155$ |
5.548292329 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 159 a - 3016\) , \( 5039 a - 64248\bigr] \) |
${y}^2={x}^3+{x}^2+\left(159a-3016\right){x}+5039a-64248$ |
| 810.1-a1 |
810.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
810.1 |
\( 2 \cdot 3^{4} \cdot 5 \) |
\( 2^{26} \cdot 3^{12} \cdot 5^{7} \) |
$2.65424$ |
$(2,a), (5,a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$0.461893610$ |
1.161419578 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -579 a + 3216\) , \( -19480 a - 45780\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(-579a+3216\right){x}-19480a-45780$ |
| 1250.3-a1 |
1250.3-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1250.3 |
\( 2 \cdot 5^{4} \) |
\( 2^{26} \cdot 5^{19} \) |
$2.95833$ |
$(2,a), (5,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{4} \) |
$1.112387565$ |
$0.277136166$ |
0.885907678 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -1614 a + 8936\) , \( 92044 a + 183836\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-1614a+8936\right){x}+92044a+183836$ |
| 1280.9-c1 |
1280.9-c |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1280.9 |
\( 2^{8} \cdot 5 \) |
\( 2^{50} \cdot 5^{7} \) |
$2.97592$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{4} \) |
$2.510309727$ |
$0.346420207$ |
2.499019599 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -1032 a + 5712\) , \( -44428 a - 95392\bigr] \) |
${y}^2={x}^3+a{x}^2+\left(-1032a+5712\right){x}-44428a-95392$ |
| 1600.1-a1 |
1600.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1600.1 |
\( 2^{6} \cdot 5^{2} \) |
\( 2^{44} \cdot 5^{13} \) |
$3.14666$ |
$(2,a), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{3} \) |
$5.958836148$ |
$0.219095376$ |
1.875874574 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 5318 a - 3809\) , \( -194953 a - 321409\bigr] \) |
${y}^2={x}^3+\left(-a+1\right){x}^2+\left(5318a-3809\right){x}-194953a-321409$ |
| 3200.19-h1 |
3200.19-h |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.19 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{44} \cdot 5^{13} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{4} \cdot 7 \) |
$1$ |
$0.219095376$ |
2.203638713 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 3760 a - 12407\) , \( 230613 a - 378963\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(3760a-12407\right){x}+230613a-378963$ |
| 3920.1-b1 |
3920.1-b |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3920.1 |
\( 2^{4} \cdot 5 \cdot 7^{2} \) |
\( 2^{38} \cdot 5^{7} \cdot 7^{6} \) |
$3.93678$ |
$(2,a), (5,a+1), (7,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{4} \) |
$2.098387848$ |
$0.261869062$ |
1.579098029 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -3731 a + 1424\) , \( 110494 a - 315136\bigr] \) |
${y}^2={x}^3+\left(a-1\right){x}^2+\left(-3731a+1424\right){x}+110494a-315136$ |
| 3920.3-a1 |
3920.3-a |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3920.3 |
\( 2^{4} \cdot 5 \cdot 7^{2} \) |
\( 2^{38} \cdot 5^{7} \cdot 7^{6} \) |
$3.93678$ |
$(2,a), (5,a+1), (7,a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{3} \cdot 7 \) |
$1$ |
$0.261869062$ |
1.316926017 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 2117 a + 7507\) , \( 90713 a - 369925\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(2117a+7507\right){x}+90713a-369925$ |
| 4050.1-c1 |
4050.1-c |
$4$ |
$14$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
4050.1 |
\( 2 \cdot 3^{4} \cdot 5^{2} \) |
\( 2^{26} \cdot 3^{12} \cdot 5^{13} \) |
$3.96902$ |
$(2,a), (5,a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{4} \) |
$8.889207331$ |
$0.206565102$ |
5.276660147 |
\( -\frac{780929100411181}{1280000000} a - \frac{1907478885001083}{1280000000} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -5448 a + 9729\) , \( 43063 a - 835459\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(-5448a+9729\right){x}+43063a-835459$ |
*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.
