| 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 |
| 324.2-a1 |
324.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{14} \cdot 3^{13} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$2.864946942$ |
1.589186630 |
\( -\frac{6897769}{576} a - \frac{28140137}{384} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -113 a - 153\) , \( -946 a - 1228\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-113a-153\right){x}-946a-1228$ |
| 324.2-b1 |
324.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{4} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$3.719838541$ |
1.031697584 |
\( -\frac{258635389823}{3} a - \frac{899809942759}{8} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -431 a - 579\) , \( 7123 a + 9248\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-431a-579\right){x}+7123a+9248$ |
| 324.2-b2 |
324.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{12} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 1 \) |
$1$ |
$3.719838541$ |
1.031697584 |
\( -\frac{2858765}{1728} a - \frac{10368559}{4608} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 10 a - 29\) , \( 131 a - 297\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(10a-29\right){x}+131a-297$ |
| 324.2-b3 |
324.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{20} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$3.719838541$ |
1.031697584 |
\( \frac{66858035}{157464} a + \frac{33822611}{52488} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 13\) , \( -27 a + 47\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+13{x}-27a+47$ |
| 324.2-c1 |
324.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{14} \cdot 3^{7} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \cdot 7 \) |
$0.004568930$ |
$18.21006223$ |
1.938356145 |
\( -\frac{6897769}{576} a - \frac{28140137}{384} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -22 a - 32\) , \( 76 a + 101\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-22a-32\right){x}+76a+101$ |
| 324.2-d1 |
324.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{5} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.102406998$ |
$14.84439033$ |
1.686476595 |
\( -\frac{59521345777}{36} a - \frac{51695280113}{24} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -9 a - 5\) , \( 25 a + 28\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-9a-5\right){x}+25a+28$ |
| 324.2-d2 |
324.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{15} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.034135666$ |
$14.84439033$ |
1.686476595 |
\( \frac{337033}{1458} a - \frac{414001}{243} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -a - 3\) , \( a + 4\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-a-3\right){x}+a+4$ |
| 324.2-e1 |
324.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{10} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$4.740482273$ |
1.314773223 |
\( \frac{635759836}{243} a - \frac{976217887}{162} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -6 a - 11\) , \( 15 a + 17\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a-11\right){x}+15a+17$ |
| 324.2-f1 |
324.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{11} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$1.196200464$ |
1.990597896 |
\( -\frac{59521345777}{36} a - \frac{51695280113}{24} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -45 a - 45\) , \( -246 a - 343\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-45a-45\right){x}-246a-343$ |
| 324.2-f2 |
324.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{9} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$10.76580417$ |
1.990597896 |
\( \frac{337033}{1458} a - \frac{414001}{243} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 0\) , \( -a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}-a$ |
| 324.2-g1 |
324.2-g |
$3$ |
$9$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{10} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$9$ |
\( 3 \) |
$1$ |
$0.254960647$ |
1.909260736 |
\( -\frac{258635389823}{3} a - \frac{899809942759}{8} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 966 a - 2295\) , \( 21885 a - 50660\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(966a-2295\right){x}+21885a-50660$ |
| 324.2-g2 |
324.2-g |
$3$ |
$9$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{6} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$2.294645827$ |
1.909260736 |
\( -\frac{2858765}{1728} a - \frac{10368559}{4608} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 6 a - 15\) , \( 53 a - 124\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(6a-15\right){x}+53a-124$ |
| 324.2-g3 |
324.2-g |
$3$ |
$9$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{14} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{4} \) |
$1$ |
$6.883937482$ |
1.909260736 |
\( \frac{66858035}{157464} a + \frac{33822611}{52488} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -2 a + 6\) , \( -10 a + 24\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-2a+6\right){x}-10a+24$ |
| 324.2-h1 |
324.2-h |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
324.2 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{16} \) |
$1.36693$ |
$(-a), (-a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 5 \) |
$1$ |
$1.135557154$ |
1.574734441 |
\( \frac{635759836}{243} a - \frac{976217887}{162} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -28 a - 42\) , \( -183 a - 244\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-28a-42\right){x}-183a-244$ |
*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.
