| 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 |
| 27.1-a1 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$6.265539006$ |
0.868873929 |
\( -\frac{24125}{27} a - \frac{1375}{27} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -a + 1\) , \( -2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a+1\right){x}-2$ |
| 27.1-a2 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{32} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$2.349577127$ |
0.868873929 |
\( -\frac{1794398270320625}{282429536481} a + \frac{1272952673786125}{94143178827} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -59 a - 135\) , \( 766 a + 829\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-59a-135\right){x}+766a+829$ |
| 27.1-a3 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{20} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{4} \) |
$1$ |
$0.783192375$ |
0.868873929 |
\( -\frac{16961124145384625}{6561} a + \frac{13019221158502750}{2187} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 464 a - 989\) , \( 6420 a - 15050\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(464a-989\right){x}+6420a-15050$ |
| 27.1-a4 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$18.79661701$ |
0.868873929 |
\( \frac{24125}{27} a - \frac{8500}{9} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( a\) , \( a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}+a+1$ |
| 27.1-a5 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{14} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$18.79661701$ |
0.868873929 |
\( -\frac{1567304375}{729} a + \frac{1203684625}{243} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -4 a - 15\) , \( a + 10\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-4a-15\right){x}+a+10$ |
| 27.1-a6 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{22} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{4} \) |
$1$ |
$3.132769503$ |
0.868873929 |
\( -\frac{449577713875}{531441} a + \frac{1037190880375}{531441} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 19 a - 74\) , \( 70 a - 224\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(19a-74\right){x}+70a-224$ |
| 27.1-a7 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{32} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.566384751$ |
0.868873929 |
\( \frac{1794398270320625}{282429536481} a + \frac{2024459751037750}{282429536481} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -26 a - 119\) , \( 340 a + 10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-26a-119\right){x}+340a+10$ |
| 27.1-a8 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$9.398308509$ |
0.868873929 |
\( -\frac{450190437580625}{27} a + \frac{345562524359500}{9} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -4 a - 150\) , \( -404 a + 118\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-4a-150\right){x}-404a+118$ |
| 27.1-a9 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{22} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$9.398308509$ |
0.868873929 |
\( \frac{449577713875}{531441} a + \frac{195871055500}{177147} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -84 a - 120\) , \( 546 a + 718\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-84a-120\right){x}+546a+718$ |
| 27.1-a10 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{14} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$6.265539006$ |
0.868873929 |
\( \frac{1567304375}{729} a + \frac{2043749500}{729} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -6 a - 14\) , \( -20 a - 26\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6a-14\right){x}-20a-26$ |
| 27.1-a11 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{20} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$4.699154254$ |
0.868873929 |
\( \frac{16961124145384625}{6561} a + \frac{22096539330123625}{6561} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -847 a + 1914\) , \( -19366 a + 44646\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-847a+1914\right){x}-19366a+44646$ |
| 27.1-a12 |
27.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$0.73443$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$3.132769503$ |
0.868873929 |
\( \frac{450190437580625}{27} a + \frac{586497135497875}{27} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -111 a - 194\) , \( -1130 a - 1304\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-111a-194\right){x}-1130a-1304$ |
*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.
