| 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 |
| 138384.3-CMa1 |
138384.3-CMa |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.3 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{16} \cdot 3^{6} \cdot 31^{8} \) |
$2.98518$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{yes}$ |
$-3$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1[2] |
$9$ |
\( 3^{2} \) |
$1$ |
$0.326086586$ |
3.388791209 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 4416 a - 1820\bigr] \) |
${y}^2={x}^{3}+4416a-1820$ |
| 138384.3-a1 |
138384.3-a |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.3 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{16} \cdot 3^{27} \cdot 31^{2} \) |
$2.98518$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \cdot 3 \) |
$2.016650234$ |
$0.181488271$ |
5.071422102 |
\( -\frac{6960893584}{177147} a - \frac{54789966496}{177147} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 2042 a + 2603\) , \( 90889 a - 106297\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(2042a+2603\right){x}+90889a-106297$ |
| 138384.3-a2 |
138384.3-a |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.3 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{16} \cdot 3^{9} \cdot 31^{2} \) |
$2.98518$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \cdot 3 \) |
$0.288092890$ |
$1.270417897$ |
5.071422102 |
\( \frac{48784}{9} a - \frac{34784}{9} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 2 a - 37\) , \( a + 71\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(2a-37\right){x}+a+71$ |
| 138384.3-b1 |
138384.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.3 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{16} \cdot 3^{3} \cdot 31^{8} \) |
$2.98518$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.423287482$ |
2.932621700 |
\( \frac{355344}{961} a + \frac{314016}{961} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -62 a - 149\) , \( 1569 a - 905\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-62a-149\right){x}+1569a-905$ |
| 138384.3-b2 |
138384.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
138384.3 |
\( 2^{4} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{8} \cdot 3^{3} \cdot 31^{7} \) |
$2.98518$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$0.846574964$ |
2.932621700 |
\( -\frac{63744}{31} a + \frac{157440}{31} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 3 a + 81\) , \( 296 a - 87\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(3a+81\right){x}+296a-87$ |
*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.
