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 |
242.3-a1 |
242.3-a |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2 \cdot 11^{8} \) |
$0.93248$ |
$(a), (2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B.1.1, 11B.3.1 |
$1$ |
\( 3 \) |
$1$ |
$2.651529103$ |
0.668122533 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -a - 5\) , \( a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a-5\right){x}+a+3$ |
1936.3-b1 |
1936.3-b |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
1936.3 |
\( 2^{4} \cdot 11^{2} \) |
\( 2^{13} \cdot 11^{2} \) |
$1.56825$ |
$(a), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B, 11B |
$1$ |
\( 2^{2} \) |
$0.075498423$ |
$4.397063578$ |
2.007574132 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( 1\) , \( -1\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+{x}-1$ |
3872.15-d1 |
3872.15-d |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
3872.15 |
\( 2^{5} \cdot 11^{2} \) |
\( 2^{13} \cdot 11^{2} \) |
$1.86497$ |
$(a), (-a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B, 11B |
$1$ |
\( 1 \) |
$1$ |
$4.397063578$ |
3.323867636 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -a - 1\) , \( -a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a-1\right){x}-a-1$ |
7744.3-a1 |
7744.3-a |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
7744.3 |
\( 2^{6} \cdot 11^{2} \) |
\( 2^{19} \cdot 11^{8} \) |
$2.21783$ |
$(a), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B, 11B |
$1$ |
\( 2 \) |
$1$ |
$0.937457104$ |
1.417301922 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( -19 a - 14\) , \( 47 a - 131\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(-19a-14\right){x}+47a-131$ |
7744.3-b1 |
7744.3-b |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
7744.3 |
\( 2^{6} \cdot 11^{2} \) |
\( 2^{19} \cdot 11^{2} \) |
$2.21783$ |
$(a), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B, 11B |
$1$ |
\( 2^{2} \) |
$0.152203951$ |
$3.109193473$ |
2.861835308 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( 3 a - 3\) , \( a - 6\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a-3\right){x}+a-6$ |
11858.3-a1 |
11858.3-a |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
11858.3 |
\( 2 \cdot 7^{2} \cdot 11^{2} \) |
\( 2 \cdot 7^{6} \cdot 11^{2} \) |
$2.46712$ |
$(a), (-2a+1), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B, 11B |
$1$ |
\( 2 \) |
$0.251895883$ |
$3.323867636$ |
2.531662201 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -a - 1\) , \( -2 a + 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-a-1\right){x}-2a+5$ |
15488.21-a1 |
15488.21-a |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
15488.21 |
\( 2^{7} \cdot 11^{2} \) |
\( 2^{7} \cdot 11^{2} \) |
$2.63746$ |
$(a), (-a+1), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B, 11B |
$1$ |
\( 1 \) |
$0.254094213$ |
$6.218386947$ |
2.388820344 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2 a + 1\) , \( -a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2a+1\right){x}-a$ |
15488.21-g1 |
15488.21-g |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
15488.21 |
\( 2^{7} \cdot 11^{2} \) |
\( 2^{7} \cdot 11^{8} \) |
$2.63746$ |
$(a), (-a+1), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B, 11B |
$1$ |
\( 3 \) |
$0.697253854$ |
$1.874914209$ |
5.929315373 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 4 a + 4\) , \( 13 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+4\right){x}+13a+7$ |
19602.3-b1 |
19602.3-b |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
19602.3 |
\( 2 \cdot 3^{4} \cdot 11^{2} \) |
\( 2 \cdot 3^{12} \cdot 11^{8} \) |
$2.79745$ |
$(a), (2a+1), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B.1.2, 11B |
$1$ |
\( 2 \) |
$2.073055864$ |
$0.883843034$ |
5.540221346 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 4 a - 41\) , \( -119 a - 33\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(4a-41\right){x}-119a-33$ |
29282.3-c1 |
29282.3-c |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
29282.3 |
\( 2 \cdot 11^{4} \) |
\( 2 \cdot 11^{8} \) |
$3.09270$ |
$(a), (-2a+3), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B, 11B.3.2 |
$1$ |
\( 1 \) |
$1$ |
$2.651529103$ |
2.004367600 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -3 a + 5\) , \( -4 a + 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-3a+5\right){x}-4a+5$ |
30976.15-b1 |
30976.15-b |
$4$ |
$33$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
30976.15 |
\( 2^{8} \cdot 11^{2} \) |
\( 2^{25} \cdot 11^{8} \) |
$3.13649$ |
$(a), (-a+1), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 11$ |
3B, 11B |
$1$ |
\( 2 \) |
$1$ |
$0.662882275$ |
1.002183800 |
\( \frac{263}{2} a + \frac{643}{2} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 8 a - 72\) , \( -244 a - 20\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(8a-72\right){x}-244a-20$ |
*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.