| 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 |
| 338.2-a1 |
338.2-a |
$1$ |
$1$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
338.2 |
\( 2 \cdot 13^{2} \) |
\( - 2^{15} \cdot 13^{9} \) |
$2.86917$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$1$ |
$4.532574040$ |
0.745150582 |
\( -\frac{37509}{16} a^{2} - \frac{92921}{32} a - \frac{4099}{8} \) |
\( \bigl[a\) , \( -1\) , \( a^{2} - a - 2\) , \( -22 a^{2} + 47 a - 13\) , \( 135 a^{2} - 346 a + 93\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}-{x}^{2}+\left(-22a^{2}+47a-13\right){x}+135a^{2}-346a+93$ |
| 338.2-b1 |
338.2-b |
$1$ |
$1$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
338.2 |
\( 2 \cdot 13^{2} \) |
\( - 2^{2} \cdot 13^{2} \) |
$2.86917$ |
$(a^2-a-2), (a^2-2a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.026007038$ |
$128.9443639$ |
1.653916823 |
\( \frac{2177}{2} a^{2} - \frac{1353}{2} a - 2323 \) |
\( \bigl[a^{2} - a - 1\) , \( a^{2} - a - 3\) , \( a^{2} - a - 2\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}$ |
| 338.2-c1 |
338.2-c |
$2$ |
$3$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
338.2 |
\( 2 \cdot 13^{2} \) |
\( - 2 \cdot 13^{7} \) |
$2.86917$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$16.95622935$ |
2.787586934 |
\( \frac{47519}{26} a^{2} - \frac{15283}{13} a - \frac{161671}{26} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{2} + 2\) , \( a^{2} - a - 1\) , \( -3\) , \( 3 a^{2} + a - 7\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}-3{x}+3a^{2}+a-7$ |
| 338.2-c2 |
338.2-c |
$2$ |
$3$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
338.2 |
\( 2 \cdot 13^{2} \) |
\( - 2^{3} \cdot 13^{9} \) |
$2.86917$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$5.652076451$ |
2.787586934 |
\( -\frac{936828579535}{4394} a^{2} + \frac{1669497916531}{4394} a - \frac{428712816945}{4394} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{2} + 2\) , \( a^{2} - a - 1\) , \( -75 a^{2} - 55 a + 52\) , \( 512 a^{2} + 659 a - 222\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-75a^{2}-55a+52\right){x}+512a^{2}+659a-222$ |
| 338.2-d1 |
338.2-d |
$2$ |
$7$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
338.2 |
\( 2 \cdot 13^{2} \) |
\( - 2 \cdot 13^{13} \) |
$2.86917$ |
$(a^2-a-2), (a^2-2a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 2^{2} \) |
$0.269775332$ |
$6.229134716$ |
1.657602329 |
\( \frac{2930762217049733062408935}{125497034} a^{2} - \frac{1009489594983775464711535}{62748517} a - \frac{9420406871887862637780499}{125497034} \) |
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 0\) , \( -24323343 a^{2} - 28460417 a + 11208468\) , \( 120295763573 a^{2} + 140756447359 a - 55433626406\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-24323343a^{2}-28460417a+11208468\right){x}+120295763573a^{2}+140756447359a-55433626406$ |
| 338.2-d2 |
338.2-d |
$2$ |
$7$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
338.2 |
\( 2 \cdot 13^{2} \) |
\( - 2^{7} \cdot 13^{7} \) |
$2.86917$ |
$(a^2-a-2), (a^2-2a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.1 |
$1$ |
\( 2^{2} \) |
$0.038539333$ |
$43.60394301$ |
1.657602329 |
\( -\frac{192290305}{104} a^{2} + \frac{119296865}{26} a - \frac{129845369}{104} \) |
\( \bigl[1\) , \( a\) , \( a^{2} - a - 2\) , \( 10 a - 2\) , \( 5 a^{2} - 12 a + 3\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+a{x}^{2}+\left(10a-2\right){x}+5a^{2}-12a+3$ |
| 338.2-e1 |
338.2-e |
$1$ |
$1$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
338.2 |
\( 2 \cdot 13^{2} \) |
\( - 2^{2} \cdot 13^{8} \) |
$2.86917$ |
$(a^2-a-2), (a^2-2a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$12.23084556$ |
2.010738625 |
\( \frac{2177}{2} a^{2} - \frac{1353}{2} a - 2323 \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( 3 a^{2} - 4 a - 8\) , \( 5 a^{2} - 3 a - 19\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(3a^{2}-4a-8\right){x}+5a^{2}-3a-19$ |
| 338.2-f1 |
338.2-f |
$1$ |
$1$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
338.2 |
\( 2 \cdot 13^{2} \) |
\( - 2^{15} \cdot 13^{3} \) |
$2.86917$ |
$(a^2-a-2), (a^2-2a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$0.003045380$ |
$126.0336304$ |
2.839485405 |
\( -\frac{37509}{16} a^{2} - \frac{92921}{32} a - \frac{4099}{8} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 2 a - 2\) , \( 1\) , \( -3 a^{2} + 3 a + 1\) , \( 3 a^{2} - 5 a + 1\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(-3a^{2}+3a+1\right){x}+3a^{2}-5a+1$ |
*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.
