| 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 |
| 9.2-a1 |
9.2-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{30} \) |
$2.06607$ |
$(a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.26036525$ |
0.951915430 |
\( -\frac{169564940890516}{282429536481} a^{2} - \frac{93767995435637}{282429536481} a + \frac{1032142453154629}{282429536481} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} + a + 3\) , \( a^{2} - 2\) , \( 2 a^{2} + 2 a - 6\) , \( -8 a^{2} - 11 a + 9\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(2a^{2}+2a-6\right){x}-8a^{2}-11a+9$ |
| 9.2-a2 |
9.2-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{18} \) |
$2.06607$ |
$(a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$30.52073051$ |
0.951915430 |
\( \frac{1812669625}{531441} a^{2} + \frac{2069390414}{531441} a - \frac{1917539791}{531441} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} + a + 3\) , \( a^{2} - 2\) , \( -3 a^{2} - 3 a + 9\) , \( -7 a^{2} - 6 a + 12\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-3a^{2}-3a+9\right){x}-7a^{2}-6a+12$ |
| 9.2-a3 |
9.2-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{14} \) |
$2.06607$ |
$(a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.26036525$ |
0.951915430 |
\( \frac{952377494237}{6561} a^{2} - \frac{2793434061809}{6561} a + \frac{1537874453914}{6561} \) |
\( \bigl[1\) , \( -a^{2} - a + 2\) , \( a\) , \( 29 a^{2} - 9 a - 123\) , \( 148 a^{2} - 17 a - 575\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(29a^{2}-9a-123\right){x}+148a^{2}-17a-575$ |
| 9.2-a4 |
9.2-a |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{10} \) |
$2.06607$ |
$(a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$30.52073051$ |
0.951915430 |
\( \frac{1737352}{81} a^{2} + \frac{3240461}{81} a - \frac{3315481}{81} \) |
\( \bigl[1\) , \( -a^{2} - a + 2\) , \( a\) , \( -a^{2} - 4 a - 3\) , \( 15 a^{2} + 14 a - 29\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-a^{2}-4a-3\right){x}+15a^{2}+14a-29$ |
| 9.2-b1 |
9.2-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{30} \) |
$2.06607$ |
$(a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$13.93653059$ |
0.869336893 |
\( -\frac{169564940890516}{282429536481} a^{2} - \frac{93767995435637}{282429536481} a + \frac{1032142453154629}{282429536481} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} - a + 3\) , \( a^{2} - 3\) , \( -2 a^{2} + 14 a - 9\) , \( -18 a^{2} - 32 a + 31\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-2a^{2}+14a-9\right){x}-18a^{2}-32a+31$ |
| 9.2-b2 |
9.2-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{18} \) |
$2.06607$ |
$(a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$27.87306118$ |
0.869336893 |
\( \frac{1812669625}{531441} a^{2} + \frac{2069390414}{531441} a - \frac{1917539791}{531441} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} - a + 3\) , \( a^{2} - 3\) , \( -2 a^{2} - 6 a + 6\) , \( -a^{2} - 5 a + 4\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-2a^{2}-6a+6\right){x}-a^{2}-5a+4$ |
| 9.2-b3 |
9.2-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{14} \) |
$2.06607$ |
$(a^2-3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$125.4287753$ |
0.869336893 |
\( \frac{952377494237}{6561} a^{2} - \frac{2793434061809}{6561} a + \frac{1537874453914}{6561} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 4\) , \( 0\) , \( 4 a^{2} + 4 a - 12\) , \( 21 a^{2} + 28 a - 12\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(4a^{2}+4a-12\right){x}+21a^{2}+28a-12$ |
| 9.2-b4 |
9.2-b |
$4$ |
$6$ |
3.3.257.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{10} \) |
$2.06607$ |
$(a^2-3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$250.8575506$ |
0.869336893 |
\( \frac{1737352}{81} a^{2} + \frac{3240461}{81} a - \frac{3315481}{81} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 4\) , \( 0\) , \( -a^{2} - a + 3\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(-a^{2}-a+3\right){x}$ |
*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.
