| 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-a1 |
675.5-a |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{11} \cdot 5^{10} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.085698759$ |
$0.945626245$ |
1.563787312 |
\( -\frac{3412573008482}{31640625} a - \frac{1003065641677}{10546875} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -70 a + 118\) , \( 86 a + 532\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-70a+118\right){x}+86a+532$ |
| 675.5-a2 |
675.5-a |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{11} \cdot 5^{10} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.342795038$ |
$0.945626245$ |
1.563787312 |
\( \frac{3412573008482}{31640625} a - \frac{6421769933513}{31640625} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -20 a + 148\) , \( -368 a - 104\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-20a+148\right){x}-368a-104$ |
| 675.5-a3 |
675.5-a |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{10} \cdot 5^{8} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.171397519$ |
$1.891252491$ |
1.563787312 |
\( \frac{5929741}{5625} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -5 a + 13\) , \( -5 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a+13\right){x}-5a+4$ |
| 675.5-a4 |
675.5-a |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{8} \cdot 5^{4} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.342795038$ |
$3.782504983$ |
1.563787312 |
\( \frac{205379}{75} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -2\) , \( -2 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}-2{x}-2a+4$ |
| 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 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 7 a - 3\) , \( -11 a - 9\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(7a-3\right){x}-11a-9$ |
| 675.5-b2 |
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 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$2.644418310$ |
1.594644241 |
\( \frac{84015547}{3375} a - \frac{96331873}{3375} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -3 a - 12\) , \( 2 a + 19\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-3a-12\right){x}+2a+19$ |
| 675.5-b3 |
675.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{14} \cdot 5^{8} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{5} \cdot 3^{2} \) |
$1$ |
$1.322209155$ |
1.594644241 |
\( -\frac{1217478647}{11390625} a + \frac{227748383}{3796875} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 2 a + 12\) , \( -34 a + 6\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(2a+12\right){x}-34a+6$ |
| 675.5-b4 |
675.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{14} \cdot 5^{8} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{5} \) |
$1$ |
$1.322209155$ |
1.594644241 |
\( \frac{1217478647}{11390625} a - \frac{534233498}{11390625} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -8 a + 3\) , \( -a + 64\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-8a+3\right){x}-a+64$ |
| 675.5-b5 |
675.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{13} \cdot 5^{13} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{4} \) |
$1$ |
$0.661104577$ |
1.594644241 |
\( -\frac{93926997067673}{19775390625} a + \frac{66606305361919}{6591796875} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 77 a + 63\) , \( -104 a + 706\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(77a+63\right){x}-104a+706$ |
| 675.5-b6 |
675.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{13} \cdot 5^{13} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$1$ |
$0.661104577$ |
1.594644241 |
\( \frac{93926997067673}{19775390625} a + \frac{105891919018084}{19775390625} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -88 a - 33\) , \( -448 a + 339\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-88a-33\right){x}-448a+339$ |
| 675.5-b7 |
675.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{19} \cdot 5^{7} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{4} \) |
$1$ |
$0.661104577$ |
1.594644241 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -173 a + 183\) , \( -310 a + 2242\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-173a+183\right){x}-310a+2242$ |
| 675.5-b8 |
675.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.5 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{19} \cdot 5^{7} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$1$ |
$0.661104577$ |
1.594644241 |
\( \frac{59052841710247}{332150625} a + \frac{4469076589604}{110716875} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 12 a + 297\) , \( -1192 a + 753\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(12a+297\right){x}-1192a+753$ |
*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.
