| 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 |
| 30976.1-a1 |
30976.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{16} \cdot 11^{6} \) |
$2.05331$ |
$(2), (11)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.047877851$ |
$1.228722931$ |
3.260612759 |
\( -\frac{199794688}{1331} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -77\) , \( 289\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-77{x}+289$ |
| 30976.1-a2 |
30976.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{16} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \) |
$0.047877851$ |
$3.686168794$ |
3.260612759 |
\( \frac{8192}{11} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 3\) , \( 1\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+3{x}+1$ |
| 30976.1-b1 |
30976.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{28} \cdot 11^{12} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.797234204$ |
$0.247311577$ |
3.079425787 |
\( \frac{13060888875}{7086244} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -784 a\) , \( 1920 a - 960\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}-784a{x}+1920a-960$ |
| 30976.1-b2 |
30976.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{48} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$2.695851306$ |
$0.494623155$ |
3.079425787 |
\( \frac{1108717875}{45056} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 345\) , \( -2631 a + 1488\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+345\right){x}-2631a+1488$ |
| 30976.1-b3 |
30976.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{32} \cdot 11^{6} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.898617102$ |
$0.494623155$ |
3.079425787 |
\( \frac{2714704875}{21296} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -464 a\) , \( -4736 a + 2368\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}-464a{x}-4736a+2368$ |
| 30976.1-b4 |
30976.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{36} \cdot 11^{4} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$5.391702612$ |
$0.247311577$ |
3.079425787 |
\( \frac{4406910829875}{7744} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 5465\) , \( -177735 a + 91600\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5465\right){x}-177735a+91600$ |
| 30976.1-c1 |
30976.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{24} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.605759488$ |
$2.301451730$ |
3.219596600 |
\( \frac{19683}{11} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 10 a - 9\) , \( a - 5\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(10a-9\right){x}+a-5$ |
| 30976.1-c2 |
30976.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{24} \cdot 11^{4} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.211518976$ |
$1.150725865$ |
3.219596600 |
\( \frac{19034163}{121} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 90 a - 89\) , \( 401 a - 245\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(90a-89\right){x}+401a-245$ |
| 30976.1-d1 |
30976.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{16} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$3.472701052$ |
2.004964887 |
\( \frac{54000}{11} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 5 a - 5\) , \( 4 a - 2\bigr] \) |
${y}^2={x}^{3}+\left(5a-5\right){x}+4a-2$ |
| 30976.1-d2 |
30976.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{20} \cdot 11^{4} \) |
$2.05331$ |
$(2), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.736350526$ |
2.004964887 |
\( \frac{1687500}{121} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 25 a - 25\) , \( -52 a + 26\bigr] \) |
${y}^2={x}^{3}+\left(25a-25\right){x}-52a+26$ |
| 30976.1-e1 |
30976.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{28} \cdot 11^{12} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.797234204$ |
$0.247311577$ |
3.079425787 |
\( \frac{13060888875}{7086244} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -784 a\) , \( -1920 a + 960\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}-784a{x}-1920a+960$ |
| 30976.1-e2 |
30976.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{48} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$2.695851306$ |
$0.494623155$ |
3.079425787 |
\( \frac{1108717875}{45056} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 345\) , \( 2631 a - 1488\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+345\right){x}+2631a-1488$ |
| 30976.1-e3 |
30976.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{32} \cdot 11^{6} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.898617102$ |
$0.494623155$ |
3.079425787 |
\( \frac{2714704875}{21296} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -464 a\) , \( 4736 a - 2368\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}-464a{x}+4736a-2368$ |
| 30976.1-e4 |
30976.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{36} \cdot 11^{4} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$5.391702612$ |
$0.247311577$ |
3.079425787 |
\( \frac{4406910829875}{7744} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 5465\) , \( 177735 a - 91600\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+5465\right){x}+177735a-91600$ |
| 30976.1-f1 |
30976.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{24} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.605759488$ |
$2.301451730$ |
3.219596600 |
\( \frac{19683}{11} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 9\) , \( -a - 4\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+9\right){x}-a-4$ |
| 30976.1-f2 |
30976.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{24} \cdot 11^{4} \) |
$2.05331$ |
$(2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.211518976$ |
$1.150725865$ |
3.219596600 |
\( \frac{19034163}{121} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 89\) , \( -401 a + 156\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+89\right){x}-401a+156$ |
| 30976.1-g1 |
30976.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{16} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$3.472701052$ |
2.004964887 |
\( \frac{54000}{11} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 5 a - 5\) , \( -4 a + 2\bigr] \) |
${y}^2={x}^{3}+\left(5a-5\right){x}-4a+2$ |
| 30976.1-g2 |
30976.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{20} \cdot 11^{4} \) |
$2.05331$ |
$(2), (11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.736350526$ |
2.004964887 |
\( \frac{1687500}{121} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 25 a - 25\) , \( 52 a - 26\bigr] \) |
${y}^2={x}^{3}+\left(25a-25\right){x}+52a-26$ |
| 30976.1-h1 |
30976.1-h |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{24} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.4.2 |
$25$ |
\( 1 \) |
$1$ |
$0.092577181$ |
2.672473023 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -125125\) , \( 16994227\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-125125{x}+16994227$ |
| 30976.1-h2 |
30976.1-h |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{24} \cdot 11^{10} \) |
$2.05331$ |
$(2), (11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5Cs.4.1 |
$1$ |
\( 5 \) |
$1$ |
$0.462885905$ |
2.672473023 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -165\) , \( 1427\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-165{x}+1427$ |
| 30976.1-h3 |
30976.1-h |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{24} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$2.314429529$ |
2.672473023 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -5\) , \( -13\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-5{x}-13$ |
| 30976.1-i1 |
30976.1-i |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
30976.1 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{16} \cdot 11^{2} \) |
$2.05331$ |
$(2), (11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$3.519868987$ |
4.064394614 |
\( -\frac{27648}{11} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4\) , \( -4\bigr] \) |
${y}^2={x}^{3}-4{x}-4$ |
*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.
