| 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 |
| 4.1-a1 |
4.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$0.45564$ |
$(2)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$1.142260539$ |
0.316806072 |
\( -\frac{1250637664527933}{32} a - \frac{1629300280935823}{32} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -29 a + 2\) , \( -52 a - 106\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-29a+2\right){x}-52a-106$ |
| 4.1-a2 |
4.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$0.45564$ |
$(2)$ |
$0$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$28.55651349$ |
0.316806072 |
\( -\frac{461373}{2} a - \frac{601423}{2} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -2 a - 2\) , \( 0\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-2a-2\right){x}$ |
| 4.1-a3 |
4.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{30} \) |
$0.45564$ |
$(2)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs, 5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$1.142260539$ |
0.316806072 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 75 a - 172\) , \( 507 a - 1170\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(75a-172\right){x}+507a-1170$ |
| 4.1-a4 |
4.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$0.45564$ |
$(2)$ |
$0$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$28.55651349$ |
0.316806072 |
\( \frac{461373}{2} a - 531398 \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( a - 3\) , \( -a + 1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(a-3\right){x}-a+1$ |
| 4.1-a5 |
4.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$0.45564$ |
$(2)$ |
$0$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$28.55651349$ |
0.316806072 |
\( \frac{1331}{8} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 3\) , \( -a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+3{x}-a+4$ |
| 4.1-a6 |
4.1-a |
$6$ |
$45$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$0.45564$ |
$(2)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$1.142260539$ |
0.316806072 |
\( \frac{1250637664527933}{32} a - \frac{719984486365939}{8} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 28 a - 27\) , \( 51 a - 158\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(28a-27\right){x}+51a-158$ |
| 9.1-a1 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( -\frac{24125}{27} a - \frac{1375}{27} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+{x}$ |
| 9.1-a2 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{26} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.712469082$ |
0.474953467 |
\( -\frac{1794398270320625}{282429536481} a + \frac{1272952673786125}{94143178827} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -5 a - 40\) , \( -56 a - 157\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-5a-40\right){x}-56a-157$ |
| 9.1-a3 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{14} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.712469082$ |
0.474953467 |
\( -\frac{16961124145384625}{6561} a + \frac{13019221158502750}{2187} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 265 a - 594\) , \( 3141 a - 7218\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(265a-594\right){x}+3141a-7218$ |
| 9.1-a4 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( \frac{24125}{27} a - \frac{8500}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}$ |
| 9.1-a5 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( -\frac{1567304375}{729} a + \frac{1203684625}{243} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -5\) , \( -3 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}-5{x}-3a-1$ |
| 9.1-a6 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{16} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.849876328$ |
0.474953467 |
\( -\frac{449577713875}{531441} a + \frac{1037190880375}{531441} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 15 a - 39\) , \( 54 a - 117\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(15a-39\right){x}+54a-117$ |
| 9.1-a7 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{26} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.712469082$ |
0.474953467 |
\( \frac{1794398270320625}{282429536481} a + \frac{2024459751037750}{282429536481} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a - 44\) , \( 51 a - 168\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a-44\right){x}+51a-168$ |
| 9.1-a8 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( -\frac{450190437580625}{27} a + \frac{345562524359500}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 15 a - 65\) , \( -69 a + 182\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(15a-65\right){x}-69a+182$ |
| 9.1-a9 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{16} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.849876328$ |
0.474953467 |
\( \frac{449577713875}{531441} a + \frac{195871055500}{177147} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -15 a - 25\) , \( -69 a - 88\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-15a-25\right){x}-69a-88$ |
| 9.1-a10 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( \frac{1567304375}{729} a + \frac{2043749500}{729} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4\) , \( 3 a + 1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}-4{x}+3a+1$ |
| 9.1-a11 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{14} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.712469082$ |
0.474953467 |
\( \frac{16961124145384625}{6561} a + \frac{22096539330123625}{6561} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -265 a - 330\) , \( -3406 a - 4407\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-265a-330\right){x}-3406a-4407$ |
| 9.1-a12 |
9.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$0.55805$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$27.39950531$ |
0.474953467 |
\( \frac{450190437580625}{27} a + \frac{586497135497875}{27} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -15 a - 49\) , \( 84 a + 163\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-15a-49\right){x}+84a+163$ |
| 17.1-a1 |
17.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$0.65422$ |
$(a+4)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.471623008$ |
0.685504883 |
\( -\frac{2659135311872}{4913} a - \frac{3464252149760}{4913} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -9 a + 18\) , \( 12 a - 32\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(-9a+18\right){x}+12a-32$ |
| 17.1-a2 |
17.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$0.65422$ |
$(a+4)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$22.24460707$ |
0.685504883 |
\( \frac{4096}{17} a - \frac{20480}{17} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( a - 2\) , \( -a + 1\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(a-2\right){x}-a+1$ |
| 17.2-a1 |
17.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17 \) |
$0.65422$ |
$(a-5)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$22.24460707$ |
0.685504883 |
\( -\frac{4096}{17} a - \frac{16384}{17} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-a-1\right){x}$ |
| 17.2-a2 |
17.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17^{3} \) |
$0.65422$ |
$(a-5)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.471623008$ |
0.685504883 |
\( \frac{2659135311872}{4913} a - \frac{6123387461632}{4913} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( 9 a + 9\) , \( -13 a - 20\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(9a+9\right){x}-13a-20$ |
| 23.1-a1 |
23.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$0.70557$ |
$(-3a-1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$12.99298185$ |
0.900901198 |
\( -\frac{156803}{23} a - \frac{50676}{23} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( a + 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(a+1\right){x}$ |
| 23.1-a2 |
23.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( - 23^{2} \) |
$0.70557$ |
$(-3a-1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$6.496490929$ |
0.900901198 |
\( \frac{50955500525}{529} a + \frac{66383751309}{529} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -4 a - 4\) , \( -13 a - 18\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-4a-4\right){x}-13a-18$ |
| 23.2-a1 |
23.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
23.2 |
\( 23 \) |
\( - 23^{2} \) |
$0.70557$ |
$(3a-4)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$6.496490929$ |
0.900901198 |
\( -\frac{50955500525}{529} a + \frac{117339251834}{529} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 8 a - 7\) , \( 10 a - 14\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(8a-7\right){x}+10a-14$ |
| 23.2-a2 |
23.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$0.70557$ |
$(3a-4)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$12.99298185$ |
0.900901198 |
\( \frac{156803}{23} a - \frac{207479}{23} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 3 a + 3\) , \( 2 a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a+3\right){x}+2a+2$ |
| 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 |
\( \frac{450190437580625}{27} a + \frac{586497135497875}{27} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -111 a - 194\) , \( -1130 a - 1304\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-111a-194\right){x}-1130a-1304$ |
| 27.2-a1 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{10} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$18.79661701$ |
0.868873929 |
\( -\frac{24125}{27} a - \frac{1375}{27} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -3 a + 1\) , \( -2 a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-3a+1\right){x}-2a+2$ |
| 27.2-a2 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{32} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.566384751$ |
0.868873929 |
\( -\frac{1794398270320625}{282429536481} a + \frac{1272952673786125}{94143178827} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 26 a - 145\) , \( -340 a + 350\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(26a-145\right){x}-340a+350$ |
| 27.2-a3 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{20} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$4.699154254$ |
0.868873929 |
\( -\frac{16961124145384625}{6561} a + \frac{13019221158502750}{2187} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( 849 a + 1064\) , \( 21278 a + 27823\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(849a+1064\right){x}+21278a+27823$ |
| 27.2-a4 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{10} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$6.265539006$ |
0.868873929 |
\( \frac{24125}{27} a - \frac{8500}{9} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( a\) , \( -2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+a{x}-2$ |
| 27.2-a5 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{14} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$6.265539006$ |
0.868873929 |
\( -\frac{1567304375}{729} a + \frac{1203684625}{243} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 6 a - 20\) , \( 20 a - 46\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(6a-20\right){x}+20a-46$ |
| 27.2-a6 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{22} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$9.398308509$ |
0.868873929 |
\( -\frac{449577713875}{531441} a + \frac{1037190880375}{531441} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 82 a - 204\) , \( -547 a + 1264\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(82a-204\right){x}-547a+1264$ |
| 27.2-a7 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{32} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$2.349577127$ |
0.868873929 |
\( \frac{1794398270320625}{282429536481} a + \frac{2024459751037750}{282429536481} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 57 a - 194\) , \( -767 a + 1595\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(57a-194\right){x}-767a+1595$ |
| 27.2-a8 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{10} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$3.132769503$ |
0.868873929 |
\( -\frac{450190437580625}{27} a + \frac{345562524359500}{9} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 111 a - 305\) , \( 1130 a - 2434\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(111a-305\right){x}+1130a-2434$ |
| 27.2-a9 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{22} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{4} \) |
$1$ |
$3.132769503$ |
0.868873929 |
\( \frac{449577713875}{531441} a + \frac{195871055500}{177147} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -19 a - 55\) , \( -70 a - 154\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-19a-55\right){x}-70a-154$ |
| 27.2-a10 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{14} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$18.79661701$ |
0.868873929 |
\( \frac{1567304375}{729} a + \frac{2043749500}{729} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 2 a - 19\) , \( -2 a + 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(2a-19\right){x}-2a+11$ |
| 27.2-a11 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{20} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{4} \) |
$1$ |
$0.783192375$ |
0.868873929 |
\( \frac{16961124145384625}{6561} a + \frac{22096539330123625}{6561} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -464 a - 525\) , \( -6420 a - 8630\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-464a-525\right){x}-6420a-8630$ |
| 27.2-a12 |
27.2-a |
$12$ |
$24$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{10} \) |
$0.73443$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$9.398308509$ |
0.868873929 |
\( \frac{450190437580625}{27} a + \frac{586497135497875}{27} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 2 a - 154\) , \( 403 a - 286\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(2a-154\right){x}+403a-286$ |
*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.
