| 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-a1 |
20.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{27} \cdot 5^{6} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.463053980$ |
$2.448298200$ |
0.814469976 |
\( \frac{4201360591}{8000000} a + \frac{36727033713}{8000000} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 4 a + 40\) , \( -38 a + 48\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(4a+40\right){x}-38a+48$ |
| 20.3-a2 |
20.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{17} \cdot 5^{2} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1.389161942$ |
$7.344894602$ |
0.814469976 |
\( -\frac{240871}{200} a + \frac{1256497}{200} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -a\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2-a{x}$ |
| 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 |
\( \frac{566667}{320} a + \frac{2148781}{320} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -4 a - 4\) , \( 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-4a-4\right){x}+16$ |
| 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 |
\( -\frac{55081295373}{32768000} a + \frac{423698409061}{32768000} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 11 a + 21\) , \( -5 a + 45\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(11a+21\right){x}-5a+45$ |
| 20.3-b1 |
20.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{30} \cdot 5^{3} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
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 |
\( \frac{5721159718441}{512000} a - \frac{17013218122889}{64000} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 78 a - 963\) , \( -1718 a + 12067\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(78a-963\right){x}-1718a+12067$ |
| 20.3-b2 |
20.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{29} \cdot 5^{2} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.414954535$ |
1.226687881 |
\( \frac{94333009}{12800} a - \frac{23122663}{12800} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 3 a + 29\) , \( 16 a - 58\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(3a+29\right){x}+16a-58$ |
| 20.3-b3 |
20.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.3 |
\( 2^{2} \cdot 5 \) |
\( 2^{34} \cdot 5 \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.414954535$ |
1.226687881 |
\( \frac{7985051}{1310720} a - \frac{205461507}{1310720} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -2 a - 3\) , \( -6 a + 35\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-2a-3\right){x}-6a+35$ |
| 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 |
\( -\frac{1520638086962303}{262144000000} a + \frac{87855796164087}{32768000000} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 68 a - 171\) , \( 452 a - 346\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(68a-171\right){x}+452a-346$ |
*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.
