| 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 |
| 31684.2-a1 |
31684.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
31684.2 |
\( 2^{2} \cdot 89^{2} \) |
\( 2^{12} \cdot 89^{4} \) |
$3.95407$ |
$(5a+2), (5a-7), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3^{2} \) |
$0.074965405$ |
$1.179104394$ |
3.837773143 |
\( \frac{1516876578875}{45118016} a + \frac{3606525111999}{45118016} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 47 a - 21\) , \( -141 a - 117\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(47a-21\right){x}-141a-117$ |
| 31684.2-b1 |
31684.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
31684.2 |
\( 2^{2} \cdot 89^{2} \) |
\( 2^{12} \cdot 89^{4} \) |
$3.95407$ |
$(5a+2), (5a-7), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3^{2} \) |
$0.074965405$ |
$1.179104394$ |
3.837773143 |
\( -\frac{1516876578875}{45118016} a + \frac{2561700845437}{22559008} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -49 a + 28\) , \( 140 a - 257\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-49a+28\right){x}+140a-257$ |
| 31684.2-c1 |
31684.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
31684.2 |
\( 2^{2} \cdot 89^{2} \) |
\( 2^{8} \cdot 89^{6} \) |
$3.95407$ |
$(5a+2), (5a-7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$4$ |
\( 2^{2} \) |
$1$ |
$0.551982120$ |
5.325723881 |
\( -\frac{18806241149857}{11279504} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -554\) , \( -5068\bigr] \) |
${y}^2+{x}{y}={x}^{3}-554{x}-5068$ |
| 31684.2-c2 |
31684.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
31684.2 |
\( 2^{2} \cdot 89^{2} \) |
\( 2^{24} \cdot 89^{2} \) |
$3.95407$ |
$(5a+2), (5a-7), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.655946361$ |
5.325723881 |
\( \frac{23639903}{364544} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 6\) , \( -28\bigr] \) |
${y}^2+{x}{y}={x}^{3}+6{x}-28$ |
| 31684.2-d1 |
31684.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
31684.2 |
\( 2^{2} \cdot 89^{2} \) |
\( 2^{28} \cdot 89^{2} \) |
$3.95407$ |
$(5a+2), (5a-7), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \cdot 7 \) |
$1$ |
$1.204226331$ |
10.16646121 |
\( \frac{9759185353}{1458176} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -44\) , \( 80\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-44{x}+80$ |
| 31684.2-d2 |
31684.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
31684.2 |
\( 2^{2} \cdot 89^{2} \) |
\( 2^{14} \cdot 89^{4} \) |
$3.95407$ |
$(5a+2), (5a-7), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$0.602113165$ |
10.16646121 |
\( \frac{35471840526793}{1013888} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -684\) , \( 6608\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-684{x}+6608$ |
*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.
