| 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 |
| 3375.11-a1 |
3375.11-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{13} \cdot 5^{11} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.726088001$ |
$0.837485832$ |
2.933528888 |
\( -\frac{15729561967}{1476225} a + \frac{7642730764}{1476225} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 36 a - 114\) , \( -221 a + 321\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(36a-114\right){x}-221a+321$ |
| 3375.11-a2 |
3375.11-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{8} \cdot 5^{10} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.452176003$ |
$1.674971665$ |
2.933528888 |
\( \frac{1768663}{1215} a + \frac{873299}{1215} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( a - 19\) , \( -12 a - 32\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a-19\right){x}-12a-32$ |
| 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 |
\( -\frac{84015547}{3375} a - \frac{4105442}{1125} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 37 a + 10\) , \( 23 a - 270\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(37a+10\right){x}+23a-270$ |
| 3375.11-b2 |
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 |
$1$ |
\( 2^{4} \) |
$2.513465836$ |
$1.182619820$ |
3.584939154 |
\( \frac{84015547}{3375} a - \frac{96331873}{3375} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 5 a - 67\) , \( -34 a + 262\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(5a-67\right){x}-34a+262$ |
| 3375.11-b3 |
3375.11-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{14} \cdot 5^{14} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{6} \cdot 3 \) |
$0.418910972$ |
$0.591309910$ |
3.584939154 |
\( -\frac{1217478647}{11390625} a + \frac{227748383}{3796875} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -3 a + 65\) , \( -203 a - 353\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-3a+65\right){x}-203a-353$ |
| 3375.11-b4 |
3375.11-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{14} \cdot 5^{14} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{6} \) |
$1.256732918$ |
$0.591309910$ |
3.584939154 |
\( \frac{1217478647}{11390625} a - \frac{534233498}{11390625} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -35 a - 12\) , \( -255 a + 594\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-35a-12\right){x}-255a+594$ |
| 3375.11-b5 |
3375.11-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{13} \cdot 5^{19} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$2.513465836$ |
$0.295654955$ |
3.584939154 |
\( -\frac{93926997067673}{19775390625} a + \frac{66606305361919}{6591796875} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 190 a + 663\) , \( -4485 a + 6129\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(190a+663\right){x}-4485a+6129$ |
| 3375.11-b6 |
3375.11-b |
$8$ |
$12$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{13} \cdot 5^{19} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{6} \cdot 3 \) |
$0.209455486$ |
$0.295654955$ |
3.584939154 |
\( \frac{93926997067673}{19775390625} a + \frac{105891919018084}{19775390625} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -268 a - 555\) , \( -4642 a - 2715\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-268a-555\right){x}-4642a-2715$ |
| 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 |
\( -\frac{59052841710247}{332150625} a + \frac{72460071479059}{332150625} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -900 a + 193\) , \( -11349 a + 17367\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-900a+193\right){x}-11349a+17367$ |
| 3375.11-b8 |
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 |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.837821945$ |
$0.295654955$ |
3.584939154 |
\( \frac{59052841710247}{332150625} a + \frac{4469076589604}{110716875} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -378 a + 1565\) , \( -10328 a - 7103\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-378a+1565\right){x}-10328a-7103$ |
| 3375.11-c1 |
3375.11-c |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{13} \cdot 5^{5} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 5 \) |
$0.095926906$ |
$1.872675252$ |
4.333078472 |
\( -\frac{15729561967}{1476225} a + \frac{7642730764}{1476225} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -12 a - 5\) , \( 17 a - 1\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-12a-5\right){x}+17a-1$ |
| 3375.11-c2 |
3375.11-c |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{8} \cdot 5^{4} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.191853812$ |
$3.745350504$ |
4.333078472 |
\( \frac{1768663}{1215} a + \frac{873299}{1215} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -2 a\) , \( -a + 5\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}-2a{x}-a+5$ |
| 3375.11-d1 |
3375.11-d |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{11} \cdot 5^{16} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.422896913$ |
4.080262942 |
\( -\frac{3412573008482}{31640625} a - \frac{1003065641677}{10546875} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -420 a + 295\) , \( -2449 a + 6743\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-420a+295\right){x}-2449a+6743$ |
| 3375.11-d2 |
3375.11-d |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{11} \cdot 5^{16} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.422896913$ |
4.080262942 |
\( \frac{3412573008482}{31640625} a - \frac{6421769933513}{31640625} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -280 a + 665\) , \( -2185 a - 5545\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-280a+665\right){x}-2185a-5545$ |
| 3375.11-d3 |
3375.11-d |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{10} \cdot 5^{14} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$1$ |
$0.845793826$ |
4.080262942 |
\( \frac{5929741}{5625} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -30 a + 40\) , \( -85 a + 5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-30a+40\right){x}-85a+5$ |
| 3375.11-d4 |
3375.11-d |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
3375.11 |
\( 3^{3} \cdot 5^{3} \) |
\( 3^{8} \cdot 5^{10} \) |
$2.25893$ |
$(-a), (a-1), (-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.691587653$ |
4.080262942 |
\( \frac{205379}{75} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 10 a - 15\) , \( -9 a + 13\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(10a-15\right){x}-9a+13$ |
*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.
