| 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 |
| 1331.1-a1 |
1331.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{14} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$2.097118535$ |
$2.186899014$ |
5.295671973 |
\( -\frac{100335366144}{14641} a - \frac{173784268800}{14641} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -217 a + 313\) , \( 7461 a - 12690\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-217a+313\right){x}+7461a-12690$ |
| 1331.1-b1 |
1331.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( - 11^{11} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$5.283973633$ |
1.525351799 |
\( -\frac{103598186306805}{14641} a + \frac{179437321887336}{14641} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -18 a - 158\) , \( 2237 a + 4468\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-18a-158\right){x}+2237a+4468$ |
| 1331.1-b2 |
1331.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{8} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$10.56794726$ |
1.525351799 |
\( \frac{19683}{11} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -13 a - 23\) , \( -18 a - 31\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-13a-23\right){x}-18a-31$ |
| 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-b4 |
1331.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( - 11^{11} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.283973633$ |
1.525351799 |
\( \frac{103598186306805}{14641} a + \frac{179437321887336}{14641} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -2168 a - 3758\) , \( 70587 a + 122280\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-2168a-3758\right){x}+70587a+122280$ |
| 1331.1-c1 |
1331.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{10} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$1.564634602$ |
2.710026626 |
\( -\frac{2861899776}{1331} a - \frac{4956954624}{1331} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -17 a + 26\) , \( -1951 a + 3373\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-17a+26\right){x}-1951a+3373$ |
| 1331.1-c2 |
1331.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{10} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$4.693903807$ |
2.710026626 |
\( \frac{2861899776}{1331} a - \frac{4956954624}{1331} \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( 817 a - 1414\) , \( -16259 a + 28160\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(817a-1414\right){x}-16259a+28160$ |
| 1331.1-d1 |
1331.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( - 11^{9} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.144310443$ |
$2.714572702$ |
3.360690442 |
\( -\frac{318097645568}{121} a + \frac{550949826752}{121} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 913 a - 1593\) , \( -20074 a + 34718\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(913a-1593\right){x}-20074a+34718$ |
| 1331.1-d2 |
1331.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( - 11^{9} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.072155221$ |
$2.714572702$ |
3.360690442 |
\( \frac{318097645568}{121} a + \frac{550949826752}{121} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 3182 a - 5515\) , \( -128631 a + 222787\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(3182a-5515\right){x}-128631a+222787$ |
| 1331.1-e1 |
1331.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( - 11^{9} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.714572702$ |
0.783629640 |
\( -\frac{318097645568}{121} a + \frac{550949826752}{121} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 912 a - 1595\) , \( 19391 a - 33578\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(912a-1595\right){x}+19391a-33578$ |
| 1331.1-e2 |
1331.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( - 11^{9} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.714572702$ |
0.783629640 |
\( \frac{318097645568}{121} a + \frac{550949826752}{121} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 3181 a - 5516\) , \( 131813 a - 228303\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(3181a-5516\right){x}+131813a-228303$ |
| 1331.1-f1 |
1331.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{10} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.130994861$ |
$4.693903807$ |
1.419998250 |
\( -\frac{2861899776}{1331} a - \frac{4956954624}{1331} \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( -17 a + 26\) , \( 1951 a - 3374\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(-17a+26\right){x}+1951a-3374$ |
| 1331.1-f2 |
1331.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{10} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.392984584$ |
$1.564634602$ |
1.419998250 |
\( \frac{2861899776}{1331} a - \frac{4956954624}{1331} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 817 a - 1414\) , \( 16259 a - 28161\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(817a-1414\right){x}+16259a-28161$ |
| 1331.1-g1 |
1331.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( - 11^{11} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$4.204418213$ |
$0.729021608$ |
1.769643082 |
\( -\frac{103598186306805}{14641} a + \frac{179437321887336}{14641} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -18 a - 158\) , \( -2237 a - 4468\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-18a-158\right){x}-2237a-4468$ |
| 1331.1-g2 |
1331.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{8} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.051104553$ |
$11.66434574$ |
1.769643082 |
\( \frac{19683}{11} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -13 a - 23\) , \( 18 a + 31\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-13a-23\right){x}+18a+31$ |
| 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 |
\( \frac{19034163}{121} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -133 a - 238\) , \( -1012 a - 1764\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-133a-238\right){x}-1012a-1764$ |
| 1331.1-g4 |
1331.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( - 11^{11} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$4.204418213$ |
$0.729021608$ |
1.769643082 |
\( \frac{103598186306805}{14641} a + \frac{179437321887336}{14641} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -2168 a - 3758\) , \( -70587 a - 122280\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2168a-3758\right){x}-70587a-122280$ |
| 1331.1-h1 |
1331.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1331.1 |
\( 11^{3} \) |
\( 11^{14} \) |
$1.86971$ |
$(-2a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$0.597928830$ |
0.690428742 |
\( -\frac{100335366144}{14641} a - \frac{173784268800}{14641} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( -217 a + 313\) , \( -7461 a + 12689\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(-217a+313\right){x}-7461a+12689$ |
*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.
