| 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 |
| 196.4-a1 |
196.4-a |
$2$ |
$2$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
196.4 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{8} \cdot 7^{10} \) |
$5.83087$ |
$(a^2-6), (a^2+2a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.993032254$ |
$32.37572978$ |
5.368713585 |
\( -\frac{10200536}{2401} a^{2} - \frac{331165}{2401} a + \frac{75472200}{2401} \) |
\( \bigl[a^{2} + a - 4\) , \( a^{2} - 6\) , \( a^{2} - 4\) , \( 28 a^{2} + 6 a - 187\) , \( 88 a^{2} + 17 a - 596\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-6\right){x}^{2}+\left(28a^{2}+6a-187\right){x}+88a^{2}+17a-596$ |
| 196.4-a2 |
196.4-a |
$2$ |
$2$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
196.4 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{4} \cdot 7^{14} \) |
$5.83087$ |
$(a^2-6), (a^2+2a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.496516127$ |
$32.37572978$ |
5.368713585 |
\( -\frac{2251961065}{5764801} a^{2} - \frac{3111870606}{5764801} a + \frac{25755881936}{5764801} \) |
\( \bigl[a\) , \( a + 1\) , \( a^{2} + a - 4\) , \( 41 a^{2} + 63 a - 129\) , \( 423 a^{2} + 642 a - 1344\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(41a^{2}+63a-129\right){x}+423a^{2}+642a-1344$ |
| 196.4-b1 |
196.4-b |
$1$ |
$1$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
196.4 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{8} \cdot 7^{7} \) |
$5.83087$ |
$(a^2-6), (a^2+2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \cdot 3 \) |
$0.135553470$ |
$16.05835584$ |
2.894424544 |
\( -\frac{141}{7} a^{2} + \frac{137}{7} a + \frac{1416}{7} \) |
\( \bigl[a^{2} - 4\) , \( a^{2} + a - 6\) , \( a^{2} - 4\) , \( a^{2} + 3 a - 4\) , \( -2 a - 8\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}+a-6\right){x}^{2}+\left(a^{2}+3a-4\right){x}-2a-8$ |
| 196.4-c1 |
196.4-c |
$1$ |
$1$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
196.4 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{2} \) |
$5.83087$ |
$(a^2-6), (a^2+2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.244816105$ |
$190.0823830$ |
5.156453666 |
\( 87451 a^{2} - 408544 a - 1447616 \) |
\( \bigl[a\) , \( a^{2} + a - 6\) , \( a\) , \( -5 a^{2} - 6 a + 15\) , \( 4 a^{2} + 7 a - 13\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}+a-6\right){x}^{2}+\left(-5a^{2}-6a+15\right){x}+4a^{2}+7a-13$ |
| 196.4-d1 |
196.4-d |
$2$ |
$2$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
196.4 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{8} \cdot 7^{6} \) |
$5.83087$ |
$(a^2-6), (a^2+2a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.125006044$ |
$81.37529086$ |
1.690768289 |
\( -12237 a^{2} - 1963 a + 84888 \) |
\( \bigl[a^{2} - 4\) , \( -a^{2} + 4\) , \( a\) , \( -318 a^{2} - 482 a + 1017\) , \( -2837 a^{2} - 4306 a + 9014\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-318a^{2}-482a+1017\right){x}-2837a^{2}-4306a+9014$ |
| 196.4-d2 |
196.4-d |
$2$ |
$2$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
196.4 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{4} \cdot 7^{6} \) |
$5.83087$ |
$(a^2-6), (a^2+2a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.062503022$ |
$81.37529086$ |
1.690768289 |
\( -809793037 a^{2} - 144300549 a + 5498539288 \) |
\( \bigl[a^{2} + a - 4\) , \( a^{2} - 6\) , \( a^{2} - 4\) , \( 78352 a^{2} + 118955 a - 248912\) , \( -13546907 a^{2} - 20566897 a + 43036813\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-6\right){x}^{2}+\left(78352a^{2}+118955a-248912\right){x}-13546907a^{2}-20566897a+43036813$ |
| 196.4-e1 |
196.4-e |
$1$ |
$1$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
196.4 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{8} \) |
$5.83087$ |
$(a^2-6), (a^2+2a-3)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3^{2} \) |
$0.011713181$ |
$143.7800668$ |
5.038561879 |
\( 87451 a^{2} - 408544 a - 1447616 \) |
\( \bigl[a^{2} + a - 4\) , \( -a^{2} + a + 6\) , \( a\) , \( -354 a^{2} - 535 a + 1135\) , \( 5824 a^{2} + 8844 a - 18497\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(-354a^{2}-535a+1135\right){x}+5824a^{2}+8844a-18497$ |
*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.
