| 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 |
| 405.1-a1 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( - 3^{16} \cdot 5 \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.698659487$ |
0.759663617 |
\( -\frac{152409672113485069453847362}{45} a + \frac{246604029693845863366701161}{45} \) |
\( \bigl[\phi\) , \( -\phi - 1\) , \( 0\) , \( -39285 \phi - 69660\) , \( 6747746 \phi + 7895612\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-39285\phi-69660\right){x}+6747746\phi+7895612$ |
| 405.1-a2 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{44} \cdot 5^{2} \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.849329743$ |
0.759663617 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -990\) , \( 22765\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-990{x}+22765$ |
| 405.1-a3 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.397318975$ |
0.759663617 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 0\) , \( -5\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-5$ |
| 405.1-a4 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{16} \cdot 5^{16} \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.849329743$ |
0.759663617 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 315\) , \( 1066\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+315{x}+1066$ |
| 405.1-a5 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{20} \cdot 5^{8} \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.397318975$ |
0.759663617 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -90\) , \( 175\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-90{x}+175$ |
| 405.1-a6 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.397318975$ |
0.759663617 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -45\) , \( -104\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-45{x}-104$ |
| 405.1-a7 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{28} \cdot 5^{4} \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.397318975$ |
0.759663617 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -1215\) , \( 16600\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-1215{x}+16600$ |
| 405.1-a8 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.849329743$ |
0.759663617 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -720\) , \( -7259\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-720{x}-7259$ |
| 405.1-a9 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{20} \cdot 5^{2} \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.397318975$ |
0.759663617 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -19440\) , \( 1048135\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-19440{x}+1048135$ |
| 405.1-a10 |
405.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( - 3^{16} \cdot 5 \) |
$0.89637$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.698659487$ |
0.759663617 |
\( \frac{152409672113485069453847362}{45} a + \frac{94194357580360793912853799}{45} \) |
\( \bigl[\phi + 1\) , \( 1\) , \( \phi + 1\) , \( 39285 \phi - 108946\) , \( -6708461 \phi + 14534412\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+{x}^{2}+\left(39285\phi-108946\right){x}-6708461\phi+14534412$ |
*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.
