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 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 2159 a + 6477\) , \( -112140 a + 261660\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(2159a+6477\right){x}-112140a+261660$ |
36.2-a1 |
36.2-a |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{23} \cdot 3^{9} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$2.004953371$ |
1.035353469 |
\( \frac{195836835}{512} a - \frac{122683695}{128} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 21 a + 141\) , \( 393 a - 550\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(21a+141\right){x}+393a-550$ |
36.2-a2 |
36.2-a |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{11} \cdot 3^{9} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$2.004953371$ |
1.035353469 |
\( -\frac{195836835}{512} a - \frac{294897945}{512} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -5 a - 36\) , \( 17 a + 92\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3-{x}^2+\left(-5a-36\right){x}+17a+92$ |
36.2-a3 |
36.2-a |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{21} \cdot 3^{3} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$6.014860114$ |
1.035353469 |
\( \frac{69255}{64} a - \frac{95715}{64} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a + 1\) , \( a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}+a+2$ |
36.2-a4 |
36.2-a |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{9} \cdot 3^{3} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$6.014860114$ |
1.035353469 |
\( -\frac{69255}{64} a - \frac{6615}{16} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -1\) , \( 1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3-{x}^2-{x}+1$ |
36.2-a5 |
36.2-a |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{19} \cdot 3^{9} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$2.004953371$ |
1.035353469 |
\( \frac{103889505}{262144} a + \frac{64790715}{65536} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -5 a + 9\) , \( -4 a - 5\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3-{x}^2+\left(-5a+9\right){x}-4a-5$ |
36.2-a6 |
36.2-a |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{31} \cdot 3^{9} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$2.004953371$ |
1.035353469 |
\( -\frac{103889505}{262144} a + \frac{363052365}{262144} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -24 a + 6\) , \( -45 a + 24\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-24a+6\right){x}-45a+24$ |
36.2-b1 |
36.2-b |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{11} \cdot 3^{9} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$2.004953371$ |
1.035353469 |
\( \frac{195836835}{512} a - \frac{122683695}{128} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 4 a - 41\) , \( -18 a + 109\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(4a-41\right){x}-18a+109$ |
36.2-b2 |
36.2-b |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{23} \cdot 3^{9} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$2.004953371$ |
1.035353469 |
\( -\frac{195836835}{512} a - \frac{294897945}{512} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -23 a + 164\) , \( -394 a - 156\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-23a+164\right){x}-394a-156$ |
36.2-b3 |
36.2-b |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{9} \cdot 3^{3} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$6.014860114$ |
1.035353469 |
\( \frac{69255}{64} a - \frac{95715}{64} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -a - 1\) , \( -a + 1\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-a-1\right){x}-a+1$ |
36.2-b4 |
36.2-b |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{21} \cdot 3^{3} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$6.014860114$ |
1.035353469 |
\( -\frac{69255}{64} a - \frac{6615}{16} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -3 a + 4\) , \( -2 a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-3a+4\right){x}-2a+4$ |
36.2-b5 |
36.2-b |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{31} \cdot 3^{9} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$2.004953371$ |
1.035353469 |
\( \frac{103889505}{262144} a + \frac{64790715}{65536} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 22 a - 16\) , \( 44 a - 20\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(22a-16\right){x}+44a-20$ |
36.2-b6 |
36.2-b |
$6$ |
$18$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{19} \cdot 3^{9} \) |
$0.84773$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$2.004953371$ |
1.035353469 |
\( -\frac{103889505}{262144} a + \frac{363052365}{262144} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 4 a + 4\) , \( 3 a - 9\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(4a+4\right){x}+3a-9$ |
38.2-a1 |
38.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.2 |
\( 2 \cdot 19 \) |
\( 2^{2} \cdot 19 \) |
$0.85927$ |
$(2,a), (2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.647328640$ |
$6.697069272$ |
0.559672526 |
\( \frac{12739275}{76} a - \frac{10237347}{76} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 2 a - 2\) , \( -4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+a{x}^2+\left(2a-2\right){x}-4$ |
38.2-a2 |
38.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.2 |
\( 2 \cdot 19 \) |
\( 2^{16} \cdot 19^{2} \) |
$0.85927$ |
$(2,a), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.323664320$ |
$6.697069272$ |
0.559672526 |
\( \frac{321975}{5776} a + \frac{7490853}{5776} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( a + 4\) , \( -2 a\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(a+4\right){x}-2a$ |
38.2-a3 |
38.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.2 |
\( 2 \cdot 19 \) |
\( 2^{8} \cdot 19 \) |
$0.85927$ |
$(2,a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.161832160$ |
$6.697069272$ |
0.559672526 |
\( -\frac{420795}{4864} a + \frac{11009439}{4864} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+a{x}^2-{x}$ |
38.2-a4 |
38.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.2 |
\( 2 \cdot 19 \) |
\( 2^{14} \cdot 19^{4} \) |
$0.85927$ |
$(2,a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.161832160$ |
$3.348534636$ |
0.559672526 |
\( -\frac{53581589505}{521284} a + \frac{17808923169}{521284} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 16 a + 24\) , \( -26 a + 96\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(16a+24\right){x}-26a+96$ |
38.2-b1 |
38.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.2 |
\( 2 \cdot 19 \) |
\( 2^{14} \cdot 19 \) |
$0.85927$ |
$(2,a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$6.697069272$ |
1.729175850 |
\( \frac{12739275}{76} a - \frac{10237347}{76} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -5 a - 7\) , \( 5 a + 10\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-5a-7\right){x}+5a+10$ |
38.2-b2 |
38.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.2 |
\( 2 \cdot 19 \) |
\( 2^{4} \cdot 19^{2} \) |
$0.85927$ |
$(2,a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.697069272$ |
1.729175850 |
\( \frac{321975}{5776} a + \frac{7490853}{5776} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -a - 1\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-a-1\right){x}-1$ |
38.2-b3 |
38.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.2 |
\( 2 \cdot 19 \) |
\( 2^{20} \cdot 19 \) |
$0.85927$ |
$(2,a), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.697069272$ |
1.729175850 |
\( -\frac{420795}{4864} a + \frac{11009439}{4864} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 2 a - 3\) , \( -2\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(2a-3\right){x}-2$ |
38.2-b4 |
38.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.2 |
\( 2 \cdot 19 \) |
\( 2^{2} \cdot 19^{4} \) |
$0.85927$ |
$(2,a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.348534636$ |
1.729175850 |
\( -\frac{53581589505}{521284} a + \frac{17808923169}{521284} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -6 a - 1\) , \( 6 a - 17\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-6a-1\right){x}+6a-17$ |
38.3-a1 |
38.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.3 |
\( 2 \cdot 19 \) |
\( 2^{14} \cdot 19^{4} \) |
$0.85927$ |
$(2,a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.161832160$ |
$3.348534636$ |
0.559672526 |
\( \frac{53581589505}{521284} a - \frac{8943166584}{130321} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -18 a + 40\) , \( 25 a + 70\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-18a+40\right){x}+25a+70$ |
38.3-a2 |
38.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.3 |
\( 2 \cdot 19 \) |
\( 2^{16} \cdot 19^{2} \) |
$0.85927$ |
$(2,a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.323664320$ |
$6.697069272$ |
0.559672526 |
\( -\frac{321975}{5776} a + \frac{1953207}{1444} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -3 a + 5\) , \( a - 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-3a+5\right){x}+a-2$ |
38.3-a3 |
38.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.3 |
\( 2 \cdot 19 \) |
\( 2^{8} \cdot 19 \) |
$0.85927$ |
$(2,a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.161832160$ |
$6.697069272$ |
0.559672526 |
\( \frac{420795}{4864} a + \frac{2647161}{1216} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2-{x}$ |
38.3-a4 |
38.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.3 |
\( 2 \cdot 19 \) |
\( 2^{2} \cdot 19 \) |
$0.85927$ |
$(2,a+1), (-2a+3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.647328640$ |
$6.697069272$ |
0.559672526 |
\( -\frac{12739275}{76} a + \frac{625482}{19} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -2 a\) , \( -4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2-2a{x}-4$ |
38.3-b1 |
38.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.3 |
\( 2 \cdot 19 \) |
\( 2^{2} \cdot 19^{4} \) |
$0.85927$ |
$(2,a+1), (-2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.348534636$ |
1.729175850 |
\( \frac{53581589505}{521284} a - \frac{8943166584}{130321} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 6 a - 7\) , \( -6 a - 11\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+a{x}^2+\left(6a-7\right){x}-6a-11$ |
38.3-b2 |
38.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.3 |
\( 2 \cdot 19 \) |
\( 2^{4} \cdot 19^{2} \) |
$0.85927$ |
$(2,a+1), (-2a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.697069272$ |
1.729175850 |
\( -\frac{321975}{5776} a + \frac{1953207}{1444} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( a - 2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+a{x}^2+\left(a-2\right){x}-1$ |
38.3-b3 |
38.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.3 |
\( 2 \cdot 19 \) |
\( 2^{20} \cdot 19 \) |
$0.85927$ |
$(2,a+1), (-2a+3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.697069272$ |
1.729175850 |
\( \frac{420795}{4864} a + \frac{2647161}{1216} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -a - 3\) , \( -2 a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(-a-3\right){x}-2a+6$ |
38.3-b4 |
38.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
38.3 |
\( 2 \cdot 19 \) |
\( 2^{14} \cdot 19 \) |
$0.85927$ |
$(2,a+1), (-2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$6.697069272$ |
1.729175850 |
\( -\frac{12739275}{76} a + \frac{625482}{19} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 6 a - 14\) , \( -11 a - 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(6a-14\right){x}-11a-5$ |
48.1-a1 |
48.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{23} \cdot 3 \) |
$0.91095$ |
$(2,a), (3,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.801619803$ |
$3.021529073$ |
0.702771756 |
\( \frac{38043647}{3} a - \frac{37451518}{3} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 5 a - 105\) , \( -41 a + 423\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(5a-105\right){x}-41a+423$ |
48.1-a2 |
48.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$0.91095$ |
$(2,a), (3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.225202475$ |
$6.043058147$ |
0.702771756 |
\( \frac{774151}{3} a - 404740 \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -a + 15\) , \( -6 a - 3\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a-1\right){x}^2+\left(-a+15\right){x}-6a-3$ |
*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.