| 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 |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{32} \cdot 5^{2} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$0.558925428$ |
0.288627850 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-110{x}-880$ |
| 15.1-a2 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$8.942806850$ |
0.288627850 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2$ |
| 15.1-a3 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{16} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.117850856$ |
0.288627850 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2+35{x}-28$ |
| 15.1-a4 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.235701712$ |
0.288627850 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-10{x}-10$ |
| 15.1-a5 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3 \cdot 5 \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$8.942806850$ |
0.288627850 |
\( \frac{46942}{15} a + \frac{725689}{15} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 3\) , \( -a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+3{x}-a+2$ |
| 15.1-a6 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3 \cdot 5 \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$8.942806850$ |
0.288627850 |
\( -\frac{46942}{15} a + \frac{772631}{15} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -2 a + 5\) , \( 2\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2+\left(-2a+5\right){x}+2$ |
| 15.1-a7 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$4.471403425$ |
0.288627850 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-5{x}+2$ |
| 15.1-a8 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$1.117850856$ |
0.288627850 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-135{x}-660$ |
| 15.1-a9 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$2.235701712$ |
0.288627850 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-80{x}+242$ |
| 15.1-a10 |
15.1-a |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$0.558925428$ |
0.288627850 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-2160{x}-39540$ |
| 15.1-b1 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{32} \cdot 5^{2} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.558925428$ |
1.154511400 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 109 a + 327\) , \( -2310 a + 5390\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(109a+327\right){x}-2310a+5390$ |
| 15.1-b2 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{2} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$8.942806850$ |
1.154511400 |
\( -\frac{1}{15} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -a - 3\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-a-3\right){x}$ |
| 15.1-b3 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{4} \cdot 5^{16} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$1.117850856$ |
1.154511400 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -36 a - 108\) , \( -189 a + 441\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-36a-108\right){x}-189a+441$ |
| 15.1-b4 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{8} \cdot 5^{8} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.235701712$ |
1.154511400 |
\( \frac{111284641}{50625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 9 a + 27\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(9a+27\right){x}$ |
| 15.1-b5 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5 \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$8.942806850$ |
1.154511400 |
\( \frac{46942}{15} a + \frac{725689}{15} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -2\) , \( 1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2-2{x}+1$ |
| 15.1-b6 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5 \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$8.942806850$ |
1.154511400 |
\( -\frac{46942}{15} a + \frac{772631}{15} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( a - 2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(a-2\right){x}-1$ |
| 15.1-b7 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{4} \cdot 5^{4} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$4.471403425$ |
1.154511400 |
\( \frac{13997521}{225} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 4 a + 12\) , \( 21 a - 49\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(4a+12\right){x}+21a-49$ |
| 15.1-b8 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{16} \cdot 5^{4} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.117850856$ |
1.154511400 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 134 a + 402\) , \( -1575 a + 3675\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(134a+402\right){x}-1575a+3675$ |
| 15.1-b9 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{2} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$2.235701712$ |
1.154511400 |
\( \frac{56667352321}{15} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 79 a + 237\) , \( 966 a - 2254\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(79a+237\right){x}+966a-2254$ |
| 15.1-b10 |
15.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 2^{12} \cdot 3^{8} \cdot 5^{2} \) |
$0.68109$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$0.558925428$ |
1.154511400 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 2159 a + 6477\) , \( -112140 a + 261660\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(2159a+6477\right){x}-112140a+261660$ |
*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.
