| 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 |
| 15.1-a1 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$64$ |
\( 1 \) |
$1$ |
$0.636997307$ |
1.315775981 |
\( -\frac{27637502042636079391}{15} a + 7135972342793472744 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 155 a - 680\) , \( 2486 a - 10032\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(155a-680\right){x}+2486a-10032$ |
| 15.1-a2 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{32} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$2.547989231$ |
1.315775981 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -110\) , \( 660\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-110{x}+660$ |
| 15.1-a3 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( -\frac{1}{15} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}$ |
| 15.1-a4 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{16} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$2.547989231$ |
1.315775981 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 35\) , \( 98\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+35{x}+98$ |
| 15.1-a5 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( \frac{111284641}{50625} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -10\) , \( -10\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-10{x}-10$ |
| 15.1-a6 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( \frac{13997521}{225} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5\) , \( -12\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-5{x}-12$ |
| 15.1-a7 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -135\) , \( 390\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-135{x}+390$ |
| 15.1-a8 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2^{2} \) |
$1$ |
$2.547989231$ |
1.315775981 |
\( \frac{56667352321}{15} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -80\) , \( -402\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-80{x}-402$ |
| 15.1-a9 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -2160\) , \( 35220\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-2160{x}+35220$ |
| 15.1-a10 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$64$ |
\( 1 \) |
$1$ |
$0.636997307$ |
1.315775981 |
\( \frac{27637502042636079391}{15} a + 7135972342793472744 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -155 a - 680\) , \( -2486 a - 10032\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-155a-680\right){x}-2486a-10032$ |
| 15.1-b1 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 2^{12} \cdot 3 \cdot 5 \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$64$ |
\( 1 \) |
$1$ |
$0.636997307$ |
1.315775981 |
\( -\frac{27637502042636079391}{15} a + 7135972342793472744 \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -2539 a - 9913\) , \( -145850 a - 565247\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-2539a-9913\right){x}-145850a-565247$ |
| 15.1-b2 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{32} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$2.547989231$ |
1.315775981 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -3519 a - 13633\) , \( 412186 a + 1596397\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-3519a-13633\right){x}+412186a+1596397$ |
| 15.1-b3 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( -\frac{1}{15} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( a + 7\) , \( -94 a - 363\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(a+7\right){x}-94a-363$ |
| 15.1-b4 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{4} \cdot 5^{16} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$2.547989231$ |
1.315775981 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 1121 a + 4347\) , \( 23958 a + 92793\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(1121a+4347\right){x}+23958a+92793$ |
| 15.1-b5 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{8} \cdot 5^{8} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( \frac{111284641}{50625} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -319 a - 1233\) , \( 2106 a + 8157\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-319a-1233\right){x}+2106a+8157$ |
| 15.1-b6 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{4} \cdot 5^{4} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( \frac{13997521}{225} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -159 a - 613\) , \( -2522 a - 9767\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-159a-613\right){x}-2522a-9767$ |
| 15.1-b7 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{16} \cdot 5^{4} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -4319 a - 16733\) , \( 294206 a + 1139457\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-4319a-16733\right){x}+294206a+1139457$ |
| 15.1-b8 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2^{2} \) |
$1$ |
$2.547989231$ |
1.315775981 |
\( \frac{56667352321}{15} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -2559 a - 9913\) , \( -144782 a - 560747\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-2559a-9913\right){x}-144782a-560747$ |
| 15.1-b9 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{8} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$10.19195692$ |
1.315775981 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -69119 a - 267833\) , \( 19314626 a + 74805717\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-69119a-267833\right){x}+19314626a+74805717$ |
| 15.1-b10 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 2^{12} \cdot 3 \cdot 5 \) |
$1.36219$ |
$(3,a), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$64$ |
\( 1 \) |
$1$ |
$0.636997307$ |
1.315775981 |
\( \frac{27637502042636079391}{15} a + 7135972342793472744 \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -40979 a - 158713\) , \( -8981954 a - 34786967\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-40979a-158713\right){x}-8981954a-34786967$ |
| 15.1-c1 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$4.801950709$ |
$15.69351105$ |
1.216108162 |
\( -\frac{27637502042636079391}{15} a + 7135972342793472744 \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 155 a - 680\) , \( -2176 a + 8672\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(155a-680\right){x}-2176a+8672$ |
| 15.1-c2 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{32} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$9.603901418$ |
$0.490422220$ |
1.216108162 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-110{x}-880$ |
| 15.1-c3 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.600243838$ |
$31.38702211$ |
1.216108162 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}$ |
| 15.1-c4 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{16} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$4.801950709$ |
$1.961688882$ |
1.216108162 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+35{x}-28$ |
| 15.1-c5 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$2.400975354$ |
$7.846755528$ |
1.216108162 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-10{x}-10$ |
| 15.1-c6 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.200487677$ |
$31.38702211$ |
1.216108162 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-5{x}+2$ |
| 15.1-c7 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.801950709$ |
$1.961688882$ |
1.216108162 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-135{x}-660$ |
| 15.1-c8 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.400975354$ |
$31.38702211$ |
1.216108162 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$ |
| 15.1-c9 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$9.603901418$ |
$0.490422220$ |
1.216108162 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2160{x}-39540$ |
| 15.1-c10 |
15.1-c |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$4.801950709$ |
$15.69351105$ |
1.216108162 |
\( \frac{27637502042636079391}{15} a + 7135972342793472744 \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -155 a - 680\) , \( 2176 a + 8672\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-155a-680\right){x}+2176a+8672$ |
| 15.1-d1 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 2^{12} \cdot 3 \cdot 5 \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.600625687$ |
$15.69351105$ |
2.634464464 |
\( -\frac{27637502042636079391}{15} a + 7135972342793472744 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2543 a - 9924\) , \( 138228 a + 535486\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2543a-9924\right){x}+138228a+535486$ |
| 15.1-d2 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{32} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$5.201251375$ |
$0.490422220$ |
2.634464464 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -3523 a - 13644\) , \( -422748 a - 1637318\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-3523a-13644\right){x}-422748a-1637318$ |
| 15.1-d3 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.325078210$ |
$31.38702211$ |
2.634464464 |
\( -\frac{1}{15} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -3 a - 4\) , \( 92 a + 362\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-3a-4\right){x}+92a+362$ |
| 15.1-d4 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{4} \cdot 5^{16} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.600625687$ |
$1.961688882$ |
2.634464464 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 1117 a + 4336\) , \( -20600 a - 79774\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(1117a+4336\right){x}-20600a-79774$ |
| 15.1-d5 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{8} \cdot 5^{8} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.300312843$ |
$7.846755528$ |
2.634464464 |
\( \frac{111284641}{50625} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -323 a - 1244\) , \( -3068 a - 11878\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-323a-1244\right){x}-3068a-11878$ |
| 15.1-d6 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{4} \cdot 5^{4} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.650156421$ |
$31.38702211$ |
2.634464464 |
\( \frac{13997521}{225} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -163 a - 624\) , \( 2040 a + 7906\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-163a-624\right){x}+2040a+7906$ |
| 15.1-d7 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{16} \cdot 5^{4} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$2.600625687$ |
$1.961688882$ |
2.634464464 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -4323 a - 16744\) , \( -307168 a - 1189678\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-4323a-16744\right){x}-307168a-1189678$ |
| 15.1-d8 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.300312843$ |
$31.38702211$ |
2.634464464 |
\( \frac{56667352321}{15} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2563 a - 9924\) , \( 137100 a + 530986\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2563a-9924\right){x}+137100a+530986$ |
| 15.1-d9 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{8} \cdot 5^{2} \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$5.201251375$ |
$0.490422220$ |
2.634464464 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -69123 a - 267844\) , \( -19521988 a - 75609238\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-69123a-267844\right){x}-19521988a-75609238$ |
| 15.1-d10 |
15.1-d |
$10$ |
$32$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 2^{12} \cdot 3 \cdot 5 \) |
$1.36219$ |
$(3,a), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.600625687$ |
$15.69351105$ |
2.634464464 |
\( \frac{27637502042636079391}{15} a + 7135972342793472744 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -40983 a - 158724\) , \( 8859012 a + 34310806\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-40983a-158724\right){x}+8859012a+34310806$ |
*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.
