| 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 |
| 138384.1-CMa1 |
138384.1-CMa |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.1 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{16} \cdot 3^{6} \cdot 31^{8} \) |
$2.98518$ |
$(-2a+1), (-6a+1), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{yes}$ |
$-3$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1[2] |
$9$ |
\( 3^{2} \) |
$1$ |
$0.326086586$ |
3.388791209 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -4416 a + 2596\bigr] \) |
${y}^2={x}^{3}-4416a+2596$ |
| 138384.1-a1 |
138384.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.1 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{16} \cdot 3^{27} \cdot 31^{2} \) |
$2.98518$ |
$(-2a+1), (-6a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \cdot 3 \) |
$2.016650234$ |
$0.181488271$ |
5.071422102 |
\( \frac{6960893584}{177147} a - \frac{61750860080}{177147} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2602 a - 2041\) , \( -86245 a - 18011\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2602a-2041\right){x}-86245a-18011$ |
| 138384.1-a2 |
138384.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.1 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{16} \cdot 3^{9} \cdot 31^{2} \) |
$2.98518$ |
$(-2a+1), (-6a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \cdot 3 \) |
$0.288092890$ |
$1.270417897$ |
5.071422102 |
\( -\frac{48784}{9} a + \frac{14000}{9} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 38 a - 1\) , \( -37 a + 109\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(38a-1\right){x}-37a+109$ |
| 138384.1-b1 |
138384.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.1 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{16} \cdot 3^{3} \cdot 31^{8} \) |
$2.98518$ |
$(-2a+1), (-6a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.423287482$ |
2.932621700 |
\( -\frac{355344}{961} a + \frac{669360}{961} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 150 a + 63\) , \( -1781 a + 813\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(150a+63\right){x}-1781a+813$ |
| 138384.1-b2 |
138384.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.1 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{8} \cdot 3^{3} \cdot 31^{7} \) |
$2.98518$ |
$(-2a+1), (-6a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$0.846574964$ |
2.932621700 |
\( \frac{63744}{31} a + \frac{93696}{31} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -80 a - 2\) , \( -213 a + 128\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-80a-2\right){x}-213a+128$ |
*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.
