| 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 |
| 32.3-a1 |
32.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{18} \) |
$1.60459$ |
$(a-4), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Ns |
$1$ |
\( 2^{2} \) |
$0.061979496$ |
$18.04319148$ |
1.184988031 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -1305 a - 4263\) , \( 1125365 a + 3685487\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1305a-4263\right){x}+1125365a+3685487$ |
| 32.3-b1 |
32.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( - 2^{27} \) |
$1.60459$ |
$(a-4), (a+3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$11.54638980$ |
2.294035035 |
\( -\frac{1536003}{4096} a + \frac{3288897}{2048} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 1560 a + 5125\) , \( 1618591 a + 5300764\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1560a+5125\right){x}+1618591a+5300764$ |
| 32.3-b2 |
32.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( - 2^{27} \) |
$1.60459$ |
$(a-4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$5.773194903$ |
2.294035035 |
\( \frac{1536003}{4096} a + \frac{5041791}{4096} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -3 a + 7\) , \( -2 a + 2\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a+7\right){x}-2a+2$ |
| 32.3-b3 |
32.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{24} \) |
$1.60459$ |
$(a-4), (a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$11.54638980$ |
2.294035035 |
\( \frac{2146689}{64} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 995 a - 4253\) , \( 34738 a - 148502\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(995a-4253\right){x}+34738a-148502$ |
| 32.3-b4 |
32.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{18} \) |
$1.60459$ |
$(a-4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$5.773194903$ |
2.294035035 |
\( \frac{8602523649}{8} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 15835 a - 67693\) , \( 2144522 a - 9167654\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15835a-67693\right){x}+2144522a-9167654$ |
| 32.3-c1 |
32.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{22} \) |
$1.60459$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$3.634541686$ |
0.962813613 |
\( \frac{247661905}{2048} a - \frac{529274331}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -1\) , \( 5 a + 9\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}-{x}+5a+9$ |
| 32.3-d1 |
32.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{22} \) |
$1.60459$ |
$(a-4), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \cdot 11 \) |
$0.022835390$ |
$10.61070305$ |
2.824215587 |
\( \frac{247661905}{2048} a - \frac{529274331}{1024} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 46211 a - 197547\) , \( -10459080 a + 44711703\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(46211a-197547\right){x}-10459080a+44711703$ |
| 32.3-e1 |
32.3-e |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{18} \) |
$1.60459$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$5.459189198$ |
4.338523642 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -2 a + 9\) , \( -a + 4\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a+9\right){x}-a+4$ |
| 32.3-f1 |
32.3-f |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( - 2^{27} \) |
$1.60459$ |
$(a-4), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{3} \) |
$0.605688956$ |
$5.773194903$ |
1.852628916 |
\( -\frac{1536003}{4096} a + \frac{3288897}{2048} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 2 a - 17\) , \( 15 a - 69\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a-17\right){x}+15a-69$ |
| 32.3-f2 |
32.3-f |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( - 2^{27} \) |
$1.60459$ |
$(a-4), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \) |
$0.605688956$ |
$11.54638980$ |
1.852628916 |
\( \frac{1536003}{4096} a + \frac{5041791}{4096} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -8401 a + 35905\) , \( -20189523 a + 86308535\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8401a+35905\right){x}-20189523a+86308535$ |
| 32.3-f3 |
32.3-f |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{24} \) |
$1.60459$ |
$(a-4), (a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{3} \) |
$1.211377913$ |
$11.54638980$ |
1.852628916 |
\( \frac{2146689}{64} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -188 a - 605\) , \( -2683 a - 8777\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-188a-605\right){x}-2683a-8777$ |
| 32.3-f4 |
32.3-f |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{18} \) |
$1.60459$ |
$(a-4), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$2.422755826$ |
$5.773194903$ |
1.852628916 |
\( \frac{8602523649}{8} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -2948 a - 9645\) , \( -170347 a - 557865\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2948a-9645\right){x}-170347a-557865$ |
*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.
