| 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 |
| 31.1-a1 |
31.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$0.47148$ |
$(5a-2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$51.50883971$ |
0.359928959 |
\( -\frac{106208}{31} a + \frac{51455}{31} \) |
\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( \phi\) , \( 0\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\phi{x}$ |
| 31.1-a2 |
31.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{2} \) |
$0.47148$ |
$(5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1.609651241$ |
0.359928959 |
\( -\frac{61725871986044215714}{961} a + \frac{99874558858644938523}{961} \) |
\( \bigl[\phi\) , \( -1\) , \( \phi + 1\) , \( 133 \phi - 141\) , \( 737 \phi - 764\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(133\phi-141\right){x}+737\phi-764$ |
| 31.1-a3 |
31.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( 31^{4} \) |
$0.47148$ |
$(5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$6.438604964$ |
0.359928959 |
\( -\frac{156520379364360}{923521} a + \frac{253260256463213}{923521} \) |
\( \bigl[\phi\) , \( -1\) , \( \phi + 1\) , \( -12 \phi - 21\) , \( 42 \phi + 10\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(-12\phi-21\right){x}+42\phi+10$ |
| 31.1-a4 |
31.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{8} \) |
$0.47148$ |
$(5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1.609651241$ |
0.359928959 |
\( \frac{11889611722383394}{852891037441} a - \frac{8629385062119691}{852891037441} \) |
\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( 31 \phi - 75\) , \( 141 \phi - 303\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(31\phi-75\right){x}+141\phi-303$ |
| 31.1-a5 |
31.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( 31^{2} \) |
$0.47148$ |
$(5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$25.75441985$ |
0.359928959 |
\( \frac{9029272560}{961} a + \frac{5599830233}{961} \) |
\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( \phi - 5\) , \( 3 \phi - 5\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(\phi-5\right){x}+3\phi-5$ |
| 31.1-a6 |
31.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$0.47148$ |
$(5a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$12.87720992$ |
0.359928959 |
\( \frac{6130703730739448}{31} a + \frac{3788983280553597}{31} \) |
\( \bigl[\phi + 1\) , \( -\phi + 1\) , \( \phi\) , \( -18 \phi + 15\) , \( 171 \phi - 265\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-18\phi+15\right){x}+171\phi-265$ |
| 31.2-a1 |
31.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{8} \) |
$0.47148$ |
$(5a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1.609651241$ |
0.359928959 |
\( -\frac{11889611722383394}{852891037441} a + \frac{3260226660263703}{852891037441} \) |
\( \bigl[1\) , \( -\phi - 1\) , \( \phi\) , \( -30 \phi - 45\) , \( -111 \phi - 117\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-30\phi-45\right){x}-111\phi-117$ |
| 31.2-a2 |
31.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$0.47148$ |
$(5a-3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$12.87720992$ |
0.359928959 |
\( -\frac{6130703730739448}{31} a + \frac{9919687011293045}{31} \) |
\( \bigl[\phi\) , \( 1\) , \( \phi + 1\) , \( 16 \phi - 2\) , \( -172 \phi - 94\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+{x}^{2}+\left(16\phi-2\right){x}-172\phi-94$ |
| 31.2-a3 |
31.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$0.47148$ |
$(5a-3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$51.50883971$ |
0.359928959 |
\( \frac{106208}{31} a - \frac{54753}{31} \) |
\( \bigl[1\) , \( -\phi - 1\) , \( \phi\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}$ |
| 31.2-a4 |
31.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$0.47148$ |
$(5a-3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$25.75441985$ |
0.359928959 |
\( -\frac{9029272560}{961} a + \frac{14629102793}{961} \) |
\( \bigl[1\) , \( -\phi - 1\) , \( \phi\) , \( -5\) , \( -3 \phi + 3\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}-5{x}-3\phi+3$ |
| 31.2-a5 |
31.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$0.47148$ |
$(5a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$6.438604964$ |
0.359928959 |
\( \frac{156520379364360}{923521} a + \frac{96739877098853}{923521} \) |
\( \bigl[\phi + 1\) , \( -\phi - 1\) , \( \phi\) , \( 10 \phi - 32\) , \( -43 \phi + 53\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(10\phi-32\right){x}-43\phi+53$ |
| 31.2-a6 |
31.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{2} \) |
$0.47148$ |
$(5a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1.609651241$ |
0.359928959 |
\( \frac{61725871986044215714}{961} a + \frac{38148686872600722809}{961} \) |
\( \bigl[\phi + 1\) , \( -\phi - 1\) , \( \phi\) , \( -135 \phi - 7\) , \( -738 \phi - 26\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-135\phi-7\right){x}-738\phi-26$ |
| 36.1-a1 |
36.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{2} \) |
$0.48944$ |
$(2), (3)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.1[2] |
$1$ |
\( 2 \) |
$1$ |
$44.29962169$ |
0.396227861 |
\( -\frac{24389}{12} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\phi{x}^{2}$ |
| 36.1-a2 |
36.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{10} \) |
$0.48944$ |
$(2), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.4[2] |
$1$ |
\( 2 \) |
$1$ |
$1.771984867$ |
0.396227861 |
\( -\frac{19465109}{248832} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi\) , \( -5 \phi - 5\) , \( -51 \phi - 37\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\phi{x}^{2}+\left(-5\phi-5\right){x}-51\phi-37$ |
| 36.1-a3 |
36.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{20} \) |
$0.48944$ |
$(2), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.4[2] |
$1$ |
\( 2 \) |
$1$ |
$1.771984867$ |
0.396227861 |
\( \frac{502270291349}{1889568} \) |
\( \bigl[\phi\) , \( \phi - 1\) , \( \phi\) , \( 165 \phi - 331\) , \( 1352 \phi - 2408\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(165\phi-331\right){x}+1352\phi-2408$ |
| 36.1-a4 |
36.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{4} \) |
$0.48944$ |
$(2), (3)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.1[2] |
$1$ |
\( 2 \) |
$1$ |
$44.29962169$ |
0.396227861 |
\( \frac{131872229}{18} \) |
\( \bigl[\phi\) , \( \phi - 1\) , \( \phi\) , \( 10 \phi - 21\) , \( -31 \phi + 51\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(10\phi-21\right){x}-31\phi+51$ |
| 41.1-a1 |
41.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
41.1 |
\( 41 \) |
\( 41 \) |
$0.50561$ |
$(a+6)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$46.26087846$ |
0.422214159 |
\( -\frac{176128}{41} a - \frac{110592}{41} \) |
\( \bigl[0\) , \( -\phi\) , \( \phi\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\phi{y}={x}^{3}-\phi{x}^{2}$ |
| 41.1-a2 |
41.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
41.1 |
\( 41 \) |
\( 41^{7} \) |
$0.50561$ |
$(a+6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 1 \) |
$1$ |
$0.944099560$ |
0.422214159 |
\( \frac{7215644871110656}{194754273881} a - \frac{11892928131395584}{194754273881} \) |
\( \bigl[0\) , \( -\phi\) , \( \phi\) , \( 10 \phi - 40\) , \( 31 \phi - 113\bigr] \) |
${y}^2+\phi{y}={x}^{3}-\phi{x}^{2}+\left(10\phi-40\right){x}+31\phi-113$ |
| 41.2-a1 |
41.2-a |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
41.2 |
\( 41 \) |
\( 41^{7} \) |
$0.50561$ |
$(a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 1 \) |
$1$ |
$0.944099560$ |
0.422214159 |
\( -\frac{7215644871110656}{194754273881} a - \frac{4677283260284928}{194754273881} \) |
\( \bigl[0\) , \( \phi - 1\) , \( \phi + 1\) , \( -10 \phi - 30\) , \( -32 \phi - 82\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-10\phi-30\right){x}-32\phi-82$ |
| 41.2-a2 |
41.2-a |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
41.2 |
\( 41 \) |
\( 41 \) |
$0.50561$ |
$(a-7)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$46.26087846$ |
0.422214159 |
\( \frac{176128}{41} a - \frac{286720}{41} \) |
\( \bigl[0\) , \( \phi - 1\) , \( \phi + 1\) , \( 0\) , \( -\phi\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}-\phi$ |
| 45.1-a1 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{4} \cdot 5 \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$0.122605555$ |
0.438646969 |
\( -\frac{152409672113485069453847362}{45} a + \frac{246604029693845863366701161}{45} \) |
\( \bigl[\phi\) , \( \phi + 1\) , \( 1\) , \( -4364 \phi - 7739\) , \( -255406 \phi - 296465\bigr] \) |
${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(-4364\phi-7739\right){x}-255406\phi-296465$ |
| 45.1-a2 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{32} \cdot 5^{2} \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.490422220$ |
0.438646969 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-110{x}-880$ |
| 45.1-a3 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$31.38702211$ |
0.438646969 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}$ |
| 45.1-a4 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{16} \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$1.961688882$ |
0.438646969 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+35{x}-28$ |
| 45.1-a5 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$7.846755528$ |
0.438646969 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-10{x}-10$ |
| 45.1-a6 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$31.38702211$ |
0.438646969 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-5{x}+2$ |
| 45.1-a7 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.961688882$ |
0.438646969 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-135{x}-660$ |
| 45.1-a8 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$31.38702211$ |
0.438646969 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$ |
| 45.1-a9 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$0.490422220$ |
0.438646969 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2160{x}-39540$ |
| 45.1-a10 |
45.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{4} \cdot 5 \) |
$0.51752$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$0.122605555$ |
0.438646969 |
\( \frac{152409672113485069453847362}{45} a + \frac{94194357580360793912853799}{45} \) |
\( \bigl[\phi + 1\) , \( \phi - 1\) , \( \phi + 1\) , \( 4364 \phi - 12105\) , \( 243301 \phi - 535402\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(4364\phi-12105\right){x}+243301\phi-535402$ |
| 49.1-a1 |
49.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{10} \) |
$0.52866$ |
$(7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.4[2] |
$1$ |
\( 1 \) |
$1$ |
$1.045448192$ |
0.467538645 |
\( -\frac{2887553024}{16807} \) |
\( \bigl[0\) , \( -\phi + 1\) , \( 1\) , \( -30 \phi - 29\) , \( -102 \phi - 84\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-30\phi-29\right){x}-102\phi-84$ |
| 49.1-a2 |
49.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{2} \) |
$0.52866$ |
$(7)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.1[2] |
$1$ |
\( 1 \) |
$1$ |
$26.13620482$ |
0.467538645 |
\( \frac{4096}{7} \) |
\( \bigl[0\) , \( \phi\) , \( 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}+\phi{x}^{2}+{x}$ |
| 55.1-a1 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11^{4} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$19.86707574$ |
0.493601465 |
\( -\frac{626283905886387}{73205} a + \frac{1013348626965991}{73205} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( 9 \phi - 25\) , \( -6 \phi + 44\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(9\phi-25\right){x}-6\phi+44$ |
| 55.1-a2 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5^{3} \cdot 11^{12} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.207452860$ |
0.493601465 |
\( -\frac{114278307303626907}{78460709418025} a + \frac{203603378036088236}{78460709418025} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( 54 \phi\) , \( -374 \phi - 198\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+54\phi{x}-374\phi-198$ |
| 55.1-a3 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11 \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$39.73415148$ |
0.493601465 |
\( \frac{45227}{55} a + \frac{26979}{55} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( -\phi - 1\) , \( -\phi\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-\phi-1\right){x}-\phi$ |
| 55.1-a4 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{6} \cdot 11^{6} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$4.414905721$ |
0.493601465 |
\( -\frac{1485675267531}{221445125} a + \frac{4152064659709}{221445125} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( -21 \phi - 25\) , \( -54 \phi - 58\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-21\phi-25\right){x}-54\phi-58$ |
| 55.1-a5 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{12} \cdot 11^{3} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.207452860$ |
0.493601465 |
\( -\frac{4560282420936767}{20796875} a + \frac{7378860561741612}{20796875} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( -16 \phi - 210\) , \( 1110 \phi - 534\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-16\phi-210\right){x}+1110\phi-534$ |
| 55.1-a6 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$39.73415148$ |
0.493601465 |
\( \frac{132583563}{605} a + \frac{166070482}{605} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( 4 \phi - 11\) , \( -9 \phi + 13\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(4\phi-11\right){x}-9\phi+13$ |
| 55.1-a7 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5^{3} \cdot 11^{3} \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$4.414905721$ |
0.493601465 |
\( \frac{754904381777}{33275} a + \frac{466557150454}{33275} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( -6 \phi - 1\) , \( \phi - 17\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-6\phi-1\right){x}+\phi-17$ |
| 55.1-a8 |
55.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{4} \cdot 11 \) |
$0.54414$ |
$(-2a+1), (-3a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$19.86707574$ |
0.493601465 |
\( \frac{48555143354501}{275} a + \frac{30008729421823}{275} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( -6 \phi - 26\) , \( 28 \phi + 8\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-6\phi-26\right){x}+28\phi+8$ |
| 55.2-a1 |
55.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11 \) |
$0.54414$ |
$(-2a+1), (-3a+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$39.73415148$ |
0.493601465 |
\( -\frac{45227}{55} a + \frac{72206}{55} \) |
\( \bigl[\phi\) , \( -\phi + 1\) , \( \phi\) , \( -\phi\) , \( 0\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}-\phi{x}$ |
| 55.2-a2 |
55.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( - 5^{3} \cdot 11^{3} \) |
$0.54414$ |
$(-2a+1), (-3a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$4.414905721$ |
0.493601465 |
\( -\frac{754904381777}{33275} a + \frac{1221461532231}{33275} \) |
\( \bigl[\phi\) , \( -\phi + 1\) , \( \phi\) , \( 4 \phi - 5\) , \( -2 \phi - 15\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(4\phi-5\right){x}-2\phi-15$ |
| 55.2-a3 |
55.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{4} \cdot 11 \) |
$0.54414$ |
$(-2a+1), (-3a+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$19.86707574$ |
0.493601465 |
\( -\frac{48555143354501}{275} a + \frac{78563872776324}{275} \) |
\( \bigl[\phi\) , \( -\phi + 1\) , \( \phi\) , \( 4 \phi - 30\) , \( -29 \phi + 37\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(4\phi-30\right){x}-29\phi+37$ |
| 55.2-a4 |
55.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( - 5^{3} \cdot 11^{12} \) |
$0.54414$ |
$(-2a+1), (-3a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.207452860$ |
0.493601465 |
\( \frac{114278307303626907}{78460709418025} a + \frac{89325070732461329}{78460709418025} \) |
\( \bigl[1\) , \( \phi\) , \( 1\) , \( -54 \phi + 54\) , \( 374 \phi - 572\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\phi{x}^{2}+\left(-54\phi+54\right){x}+374\phi-572$ |
| 55.2-a5 |
55.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{6} \cdot 11^{6} \) |
$0.54414$ |
$(-2a+1), (-3a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$4.414905721$ |
0.493601465 |
\( \frac{1485675267531}{221445125} a + \frac{2666389392178}{221445125} \) |
\( \bigl[1\) , \( \phi\) , \( 1\) , \( 21 \phi - 46\) , \( 54 \phi - 112\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\phi{x}^{2}+\left(21\phi-46\right){x}+54\phi-112$ |
| 55.2-a6 |
55.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$0.54414$ |
$(-2a+1), (-3a+2)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$39.73415148$ |
0.493601465 |
\( -\frac{132583563}{605} a + \frac{59730809}{121} \) |
\( \bigl[\phi\) , \( -\phi + 1\) , \( \phi\) , \( -6 \phi - 5\) , \( 8 \phi + 5\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-6\phi-5\right){x}+8\phi+5$ |
| 55.2-a7 |
55.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{12} \cdot 11^{3} \) |
$0.54414$ |
$(-2a+1), (-3a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.207452860$ |
0.493601465 |
\( \frac{4560282420936767}{20796875} a + \frac{563715628160969}{4159375} \) |
\( \bigl[1\) , \( \phi\) , \( 1\) , \( 16 \phi - 226\) , \( -1110 \phi + 576\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\phi{x}^{2}+\left(16\phi-226\right){x}-1110\phi+576$ |
| 55.2-a8 |
55.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11^{4} \) |
$0.54414$ |
$(-2a+1), (-3a+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$19.86707574$ |
0.493601465 |
\( \frac{626283905886387}{73205} a + \frac{387064721079604}{73205} \) |
\( \bigl[1\) , \( \phi\) , \( 1\) , \( -9 \phi - 16\) , \( 6 \phi + 38\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\phi{x}^{2}+\left(-9\phi-16\right){x}+6\phi+38$ |
| 64.1-a1 |
64.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
64.1 |
\( 2^{6} \) |
\( - 2^{20} \) |
$0.56516$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.397417474$ |
0.536078844 |
\( -2711191688 a + 4386800300 \) |
\( \bigl[0\) , \( \phi - 1\) , \( 0\) , \( 14 \phi - 25\) , \( \phi - 59\bigr] \) |
${y}^2={x}^{3}+\left(\phi-1\right){x}^{2}+\left(14\phi-25\right){x}+\phi-59$ |
| 64.1-a2 |
64.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
64.1 |
\( 2^{6} \) |
\( - 2^{16} \) |
$0.56516$ |
$(2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.17933979$ |
0.536078844 |
\( -548896 a + 889584 \) |
\( \bigl[0\) , \( \phi - 1\) , \( 0\) , \( 4 \phi\) , \( 4\bigr] \) |
${y}^2={x}^{3}+\left(\phi-1\right){x}^{2}+4\phi{x}+4$ |
*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.
