| 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-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 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -173 a + 183\) , \( -310 a + 2242\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-173a+183\right){x}-310a+2242$ |
| 675.8-b7 |
675.8-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
675.8 |
\( 3^{3} \cdot 5^{2} \) |
\( 3^{19} \cdot 5^{7} \) |
$1.51064$ |
$(-a), (a-1), (-a-1), (a-2)$ |
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 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( -14 a + 310\) , \( 1191 a - 438\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-14a+310\right){x}+1191a-438$ |
| 3375.10-b7 |
3375.10-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.10 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{19} \cdot 5^{13} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{6} \cdot 3 \) |
$0.258709874$ |
$0.295654955$ |
4.427953515 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 912 a - 500\) , \( -10094 a - 9468\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(912a-500\right){x}-10094a-9468$ |
| 3375.11-b7 |
3375.11-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{19} \cdot 5^{13} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{6} \) |
$0.628366459$ |
$0.295654955$ |
3.584939154 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -900 a + 193\) , \( -11349 a + 17367\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-900a+193\right){x}-11349a+17367$ |
| 3375.6-b7 |
3375.6-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.6 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{19} \cdot 5^{13} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.837821945$ |
$0.295654955$ |
3.584939154 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 378 a + 1188\) , \( 11515 a - 19753\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(378a+1188\right){x}+11515a-19753$ |
| 3375.7-b7 |
3375.7-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.7 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{19} \cdot 5^{13} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$3.104518489$ |
$0.295654955$ |
4.427953515 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -208 a - 1373\) , \( -5354 a - 18039\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-208a-1373\right){x}-5354a-18039$ |
| 16875.13-f8 |
16875.13-f |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
16875.13 |
\( 3^{3} \cdot 5^{4} \) |
\( 3^{19} \cdot 5^{19} \) |
$3.37789$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.132220915$ |
1.275715392 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -326 a + 7709\) , \( 155964 a - 46151\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-326a+7709\right){x}+155964a-46151$ |
| 16875.8-f8 |
16875.8-f |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
16875.8 |
\( 3^{3} \cdot 5^{4} \) |
\( 3^{19} \cdot 5^{19} \) |
$3.37789$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.132220915$ |
1.275715392 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -4320 a + 4594\) , \( -43608 a + 258905\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4320a+4594\right){x}-43608a+258905$ |
| 27225.5-b8 |
27225.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.5 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{13} \cdot 5^{7} \cdot 11^{6} \) |
$3.80695$ |
$(-a), (a-1), (-a-1), (a-2), (-2a+1)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$3.048265448$ |
$0.345250605$ |
5.077043362 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -409 a + 1085\) , \( -4644 a - 10458\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-409a+1085\right){x}-4644a-10458$ |
*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.
