| 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-b4 |
20.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{39} \cdot 5^{6} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.138318178$ |
1.226687881 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 68 a - 171\) , \( 452 a - 346\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(68a-171\right){x}+452a-346$ |
| 100.4-b4 |
100.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
100.4 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{39} \cdot 5^{12} \) |
$1.57333$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \cdot 3^{2} \) |
$0.161196103$ |
$0.509071365$ |
4.244678959 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 427 a - 226\) , \( 3712 a + 7964\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+a{x}^2+\left(427a-226\right){x}+3712a+7964$ |
| 500.7-b4 |
500.7-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
500.7 |
\( 2^{2} \cdot 5^{3} \) |
\( 2^{27} \cdot 5^{12} \) |
$2.35267$ |
$(2,a), (2,a+1), (5,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.509071365$ |
1.097182995 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -42 a - 254\) , \( 264 a + 1500\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-42a-254\right){x}+264a+1500$ |
| 640.13-i4 |
640.13-i |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.13 |
\( 2^{7} \cdot 5 \) |
\( 2^{57} \cdot 5^{6} \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1.192340186$ |
$0.402456251$ |
6.205410396 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -374 a + 1781\) , \( -7461 a - 23098\bigr] \) |
${y}^2={x}^3-a{x}^2+\left(-374a+1781\right){x}-7461a-23098$ |
| 640.3-d4 |
640.3-d |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
640.3 |
\( 2^{7} \cdot 5 \) |
\( 2^{57} \cdot 5^{6} \) |
$2.50244$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$2.309693430$ |
$0.402456251$ |
2.671235346 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -659 a + 856\) , \( -4147 a + 33368\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-659a+856\right){x}-4147a+33368$ |
| 1280.9-f4 |
1280.9-f |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1280.9 |
\( 2^{8} \cdot 5 \) |
\( 2^{63} \cdot 5^{6} \) |
$2.97592$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \) |
$2.975311498$ |
$0.284579544$ |
4.866371412 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 1102 a - 2821\) , \( -32241 a + 30717\bigr] \) |
${y}^2={x}^3+\left(-a-1\right){x}^2+\left(1102a-2821\right){x}-32241a+30717$ |
| 1620.3-a4 |
1620.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1620.3 |
\( 2^{2} \cdot 3^{4} \cdot 5 \) |
\( 2^{39} \cdot 3^{12} \cdot 5^{6} \) |
$3.15644$ |
$(2,a), (2,a+1), (5,a+1), (3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{5} \cdot 3^{2} \) |
$2.577321044$ |
$0.379439392$ |
5.620566207 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 621 a - 1584\) , \( -14067 a + 14148\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(621a-1584\right){x}-14067a+14148$ |
| 2500.8-d4 |
2500.8-d |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
2500.8 |
\( 2^{2} \cdot 5^{4} \) |
\( 2^{39} \cdot 5^{18} \) |
$3.51807$ |
$(2,a), (2,a+1), (5,a+1), (5,a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{6} \cdot 3 \) |
$1$ |
$0.227663635$ |
3.925401220 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 1720 a - 4408\) , \( 61680 a - 56688\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a+1\right){x}^2+\left(1720a-4408\right){x}+61680a-56688$ |
| 3200.19-i4 |
3200.19-i |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.19 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{57} \cdot 5^{12} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$0.179983907$ |
6.982429830 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -3209 a + 4954\) , \( 42045 a - 352139\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-3209a+4954\right){x}+42045a-352139$ |
| 3200.4-d4 |
3200.4-d |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3200.4 |
\( 2^{7} \cdot 5^{2} \) |
\( 2^{57} \cdot 5^{12} \) |
$3.74203$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{7} \cdot 3 \) |
$1$ |
$0.179983907$ |
6.206604293 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -3215 a - 1673\) , \( -127503 a + 135895\bigr] \) |
${y}^2={x}^3+{x}^2+\left(-3215a-1673\right){x}-127503a+135895$ |
*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.
