| 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 |
| 675.5-b1 |
675.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{10} \cdot 5^{4} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$2.644418310$ |
1.594644241 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 7 a - 3\) , \( -11 a - 9\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(7a-3\right){x}-11a-9$ |
| 675.8-b1 |
675.8-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.8 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{10} \cdot 5^{4} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$2.644418310$ |
1.594644241 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 3 a - 15\) , \( -2 a + 21\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(3a-15\right){x}-2a+21$ |
| 3375.10-b1 |
3375.10-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.10 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{10} \cdot 5^{10} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$3.104518489$ |
$1.182619820$ |
4.427953515 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -39 a + 1\) , \( 101 a - 91\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-39a+1\right){x}+101a-91$ |
| 3375.11-b1 |
3375.11-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{10} \cdot 5^{10} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.837821945$ |
$1.182619820$ |
3.584939154 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 37 a + 10\) , \( 23 a - 270\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(37a+10\right){x}+23a-270$ |
| 3375.6-b1 |
3375.6-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.6 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{10} \cdot 5^{10} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$2.513465836$ |
$1.182619820$ |
3.584939154 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -7 a - 62\) , \( 33 a + 228\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-7a-62\right){x}+33a+228$ |
| 3375.7-b1 |
3375.7-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.7 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{10} \cdot 5^{10} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.034839496$ |
$1.182619820$ |
4.427953515 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -3 a + 67\) , \( 150 a - 56\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-3a+67\right){x}+150a-56$ |
| 16875.13-f2 |
16875.13-f |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
16875.13 |
\( 3^{3} \cdot 5^{4} \) |
\( 3^{10} \cdot 5^{16} \) |
$3.37789$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.528883662$ |
1.275715392 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 72 a - 351\) , \( -821 a + 2196\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(72a-351\right){x}-821a+2196$ |
| 16875.8-f2 |
16875.8-f |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
16875.8 |
\( 3^{3} \cdot 5^{4} \) |
\( 3^{10} \cdot 5^{16} \) |
$3.37789$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.528883662$ |
1.275715392 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 202 a - 117\) , \( -1080 a - 1303\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(202a-117\right){x}-1080a-1303$ |
| 27225.5-b2 |
27225.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.5 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{4} \cdot 5^{4} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (a-2), (-2a+1)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$3.048265448$ |
$1.381002422$ |
5.077043362 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 24 a - 42\) , \( 80 a - 84\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(24a-42\right){x}+80a-84$ |
*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.
