| 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 |
| 20.4-a1 |
20.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{17} \cdot 5^{2} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1.389161942$ |
$7.344894602$ |
0.814469976 |
\( \frac{240871}{200} a + \frac{507813}{100} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -a - 1\) , \( -a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}-a$ |
| 20.4-a2 |
20.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{27} \cdot 5^{6} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.463053980$ |
$2.448298200$ |
0.814469976 |
\( -\frac{4201360591}{8000000} a + \frac{639506161}{125000} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -6 a + 44\) , \( 37 a + 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-6a+44\right){x}+37a+10$ |
| 20.4-a3 |
20.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{19} \cdot 5 \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$2.778323884$ |
$7.344894602$ |
0.814469976 |
\( -\frac{566667}{320} a + \frac{339431}{40} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( a\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+a{x}$ |
| 20.4-a4 |
20.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{33} \cdot 5^{3} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$0.926107961$ |
$2.448298200$ |
0.814469976 |
\( \frac{55081295373}{32768000} a + \frac{46077139211}{4096000} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -14 a + 40\) , \( 30 a + 144\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-14a+40\right){x}+30a+144$ |
| 20.4-b1 |
20.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{30} \cdot 5^{3} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
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 |
\( -\frac{5721159718441}{512000} a - \frac{130384585264671}{512000} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -80 a - 883\) , \( 1717 a + 10350\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2+\left(-80a-883\right){x}+1717a+10350$ |
| 20.4-b2 |
20.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{39} \cdot 5^{6} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
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 |
\( \frac{1520638086962303}{262144000000} a - \frac{817791717649607}{262144000000} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -70 a - 103\) , \( -453 a + 106\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-70a-103\right){x}-453a+106$ |
| 20.4-b3 |
20.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{34} \cdot 5 \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
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 |
\( -\frac{7985051}{1310720} a - \frac{24684557}{163840} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -3\) , \( 5 a + 30\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2-3{x}+5a+30$ |
| 20.4-b4 |
20.4-b |
$4$ |
$6$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
20.4 |
\( 2^{2} \cdot 5 \) |
\( 2^{29} \cdot 5^{2} \) |
$1.05215$ |
$(2,a), (2,a+1), (5,a+3)$ |
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 |
\( -\frac{94333009}{12800} a + \frac{35605173}{6400} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -5 a + 32\) , \( -17 a - 42\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-5a+32\right){x}-17a-42$ |
*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.
