| 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 |
| 891.3-a1 |
891.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{6} \cdot 11^{2} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.075642846$ |
$5.314975105$ |
0.969756591 |
\( \frac{19683}{11} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -2\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-2{x}$ |
| 891.3-a2 |
891.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{6} \cdot 11^{4} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.151285692$ |
$2.657487552$ |
0.969756591 |
\( \frac{19034163}{121} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -17\) , \( 30\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-17{x}+30$ |
| 891.3-b1 |
891.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{36} \cdot 11^{2} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$2.957701004$ |
$0.341861992$ |
2.438926612 |
\( \frac{9090072503}{5845851} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 391\) , \( -1092\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+391{x}-1092$ |
| 891.3-b2 |
891.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{24} \cdot 11^{4} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.478850502$ |
$0.683723984$ |
2.438926612 |
\( \frac{169112377}{88209} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -104\) , \( -102\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-104{x}-102$ |
| 891.3-b3 |
891.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{18} \cdot 11^{2} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.739425251$ |
$1.367447969$ |
2.438926612 |
\( \frac{30664297}{297} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -59\) , \( 186\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-59{x}+186$ |
| 891.3-b4 |
891.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{42} \cdot 11 \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$5.915402008$ |
$0.170930996$ |
2.438926612 |
\( -\frac{450360153235512010}{3106724901291} a + \frac{211862156595042847}{1035574967097} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -450 a + 4576\) , \( -65628 a + 3012\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-450a+4576\right){x}-65628a+3012$ |
| 891.3-b5 |
891.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{42} \cdot 11 \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$5.915402008$ |
$0.170930996$ |
2.438926612 |
\( \frac{450360153235512010}{3106724901291} a + \frac{185226316549616531}{3106724901291} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 450 a + 4126\) , \( 65628 a - 62616\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(450a+4126\right){x}+65628a-62616$ |
| 891.3-b6 |
891.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{18} \cdot 11^{8} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.957701004$ |
$0.341861992$ |
2.438926612 |
\( \frac{347873904937}{395307} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1319\) , \( -18084\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-1319{x}-18084$ |
| 891.3-c1 |
891.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{18} \cdot 11^{2} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.771658368$ |
2.136700387 |
\( \frac{19683}{11} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -15\) , \( 8\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-15{x}+8$ |
| 891.3-c2 |
891.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{18} \cdot 11^{4} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.885829184$ |
2.136700387 |
\( \frac{19034163}{121} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -150\) , \( -667\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-150{x}-667$ |
| 891.3-d1 |
891.3-d |
$3$ |
$25$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{12} \cdot 11^{2} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 5$ |
2Cn, 5B.4.2 |
$1$ |
\( 2 \) |
$7.655751664$ |
$0.123436241$ |
2.279419039 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -70383\) , \( 7187035\bigr] \) |
${y}^2+{y}={x}^{3}-70383{x}+7187035$ |
| 891.3-d2 |
891.3-d |
$3$ |
$25$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{12} \cdot 11^{10} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 5$ |
2Cn, 5Cs.4.1 |
$1$ |
\( 2 \) |
$1.531150332$ |
$0.617181207$ |
2.279419039 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -93\) , \( 625\bigr] \) |
${y}^2+{y}={x}^{3}-93{x}+625$ |
| 891.3-d3 |
891.3-d |
$3$ |
$25$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{12} \cdot 11^{2} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 5$ |
2Cn, 5B.4.1 |
$1$ |
\( 2 \) |
$0.306230066$ |
$3.085906039$ |
2.279419039 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -3\) , \( -5\bigr] \) |
${y}^2+{y}={x}^{3}-3{x}-5$ |
| 891.3-e1 |
891.3-e |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{9} \cdot 11^{2} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.452601877$ |
2.957949159 |
\( \frac{393194}{11} a - \frac{16965365}{11} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( 16 a\) , \( -19 a - 36\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+16a{x}-19a-36$ |
| 891.3-e2 |
891.3-e |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{9} \cdot 11 \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.905203755$ |
2.957949159 |
\( \frac{7136}{11} a + \frac{4759}{11} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( a\) , \( -a\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+a{x}-a$ |
| 891.3-f1 |
891.3-f |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{9} \cdot 11^{2} \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.452601877$ |
2.957949159 |
\( -\frac{393194}{11} a - 1506561 \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( -17 a + 16\) , \( 19 a - 55\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(-17a+16\right){x}+19a-55$ |
| 891.3-f2 |
891.3-f |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
891.3 |
\( 3^{4} \cdot 11 \) |
\( 3^{9} \cdot 11 \) |
$1.61921$ |
$(-a), (a-1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.905203755$ |
2.957949159 |
\( -\frac{7136}{11} a + \frac{11895}{11} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( -2 a + 1\) , \( a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(-2a+1\right){x}+a-1$ |
*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.
