| 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 |
| 1156.1-a1 |
1156.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{2} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$5.558309054$ |
$20.21098874$ |
5.143645881 |
\( \frac{3048625}{1088} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-3{x}+1$ |
| 1156.1-a2 |
1156.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{2} \cdot 17^{12} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$8.337463582$ |
$2.245665415$ |
5.143645881 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \) |
${y}^2+{x}{y}={x}^{3}-113{x}-329$ |
| 1156.1-a3 |
1156.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{4} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$2.779154527$ |
$20.21098874$ |
5.143645881 |
\( \frac{8805624625}{2312} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \) |
${y}^2+{x}{y}={x}^{3}-43{x}+105$ |
| 1156.1-a4 |
1156.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{4} \cdot 17^{6} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$16.67492716$ |
$2.245665415$ |
5.143645881 |
\( \frac{120920208625}{19652} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \) |
${y}^2+{x}{y}={x}^{3}-103{x}-411$ |
| 1156.1-b1 |
1156.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{10} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$49$ |
\( 2 \) |
$1$ |
$0.193536546$ |
2.605260340 |
\( -\frac{71975722083173}{90870848} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -6066 a - 19059\) , \( -502179 a - 1576915\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-6066a-19059\right){x}-502179a-1576915$ |
| 1156.1-c1 |
1156.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{2} \cdot 17^{6} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$0.494304984$ |
$6.285111710$ |
6.827945383 |
\( -\frac{1839311441}{83521} a + \frac{15268131435}{167042} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 8 a + 24\) , \( 52 a + 162\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(8a+24\right){x}+52a+162$ |
| 1156.1-c2 |
1156.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{4} \cdot 17^{3} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.988609968$ |
$25.14044684$ |
6.827945383 |
\( \frac{248918}{289} a + \frac{5190061}{1156} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -2 a - 6\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-2a-6\right){x}$ |
| 1156.1-d1 |
1156.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{5} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$6.530228532$ |
3.587983496 |
\( \frac{482771929}{83521} a - \frac{16553184881}{668168} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 49 a - 198\) , \( -307 a + 1269\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(49a-198\right){x}-307a+1269$ |
| 1156.1-e1 |
1156.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{4} \cdot 17^{3} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.988609968$ |
$25.14044684$ |
6.827945383 |
\( -\frac{248918}{289} a + \frac{6185733}{1156} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 2 a - 8\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a-8\right){x}$ |
| 1156.1-e2 |
1156.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{2} \cdot 17^{6} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$0.494304984$ |
$6.285111710$ |
6.827945383 |
\( \frac{1839311441}{83521} a + \frac{11589508553}{167042} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -8 a + 32\) , \( -52 a + 214\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a+32\right){x}-52a+214$ |
| 1156.1-f1 |
1156.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{5} \) |
$3.79329$ |
$(a+5), (a-6), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$6.530228532$ |
3.587983496 |
\( -\frac{482771929}{83521} a - \frac{12691009449}{668168} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -49 a - 149\) , \( 307 a + 962\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-49a-149\right){x}+307a+962$ |
*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.
