| 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 |
| 1089.1-j3 |
1089.1-j |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{4} \) |
$1.77822$ |
$(a), (-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.151285692$ |
$20.23607837$ |
1.767516929 |
\( \frac{19034163}{121} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -17\) , \( 30\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-17{x}+30$ |
| 1089.1-k3 |
1089.1-k |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.1 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{4} \) |
$1.77822$ |
$(a), (-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.037460680$ |
$5.583880404$ |
3.344622652 |
\( \frac{19034163}{121} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -18\) , \( -31\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-18{x}-31$ |
| 1331.1-b3 |
1331.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{10} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.56794726$ |
1.525351799 |
\( \frac{19034163}{121} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -133 a - 238\) , \( 1012 a + 1764\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-133a-238\right){x}+1012a+1764$ |
| 1331.1-g3 |
1331.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{10} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.102209106$ |
$2.916086435$ |
1.769643082 |
\( \frac{19034163}{121} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -133 a - 238\) , \( -1012 a - 1764\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-133a-238\right){x}-1012a-1764$ |
| 1331.2-b3 |
1331.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.2 |
\( 11^{3} \) |
\( 11^{10} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.56794726$ |
1.525351799 |
\( \frac{19034163}{121} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 133 a - 238\) , \( -1012 a + 1764\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(133a-238\right){x}-1012a+1764$ |
| 1331.2-g3 |
1331.2-g |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.2 |
\( 11^{3} \) |
\( 11^{10} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.102209106$ |
$2.916086435$ |
1.769643082 |
\( \frac{19034163}{121} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 133 a - 238\) , \( 1012 a - 1764\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(133a-238\right){x}+1012a-1764$ |
| 1936.1-f3 |
1936.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1936.1 |
\( 2^{4} \cdot 11^{2} \) |
\( 2^{12} \cdot 11^{4} \) |
$2.05331$ |
$(a+1), (-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.605759488$ |
$9.205806923$ |
3.219596600 |
\( \frac{19034163}{121} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -a - 24\) , \( 19 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-a-24\right){x}+19a-1$ |
| 1936.1-s3 |
1936.1-s |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1936.1 |
\( 2^{4} \cdot 11^{2} \) |
\( 2^{12} \cdot 11^{4} \) |
$2.05331$ |
$(a+1), (-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.605759488$ |
$9.205806923$ |
3.219596600 |
\( \frac{19034163}{121} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -a - 24\) , \( -20 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-24\right){x}-20a-1$ |
*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.
