| 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 |
| 24.1-a1 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.562595083$ |
$1.817673508$ |
6.019284716 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -59 a + 116\) , \( -2672 a - 19486\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-59a+116\right){x}-2672a-19486$ |
| 24.1-a2 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.125190167$ |
$7.270694035$ |
6.019284716 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( 7\bigr] \) |
${y}^2={x}^3+6{x}+7$ |
| 24.1-a3 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$6.250380335$ |
$7.270694035$ |
6.019284716 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 21 a + 81\) , \( -47 a + 412\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(21a+81\right){x}-47a+412$ |
| 24.1-a4 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$3.125190167$ |
$3.635347017$ |
6.019284716 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 101 a + 46\) , \( -1202 a - 3658\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(101a+46\right){x}-1202a-3658$ |
| 24.1-a5 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$12.50076067$ |
$3.635347017$ |
6.019284716 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 261 a - 24\) , \( 2158 a + 24154\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(261a-24\right){x}+2158a+24154$ |
| 24.1-a6 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$6.250380335$ |
$1.817673508$ |
6.019284716 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 1541 a - 584\) , \( -56012 a - 292630\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(1541a-584\right){x}-56012a-292630$ |
| 24.1-b1 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$1.817673508$ |
0.963026950 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -67 a + 79\) , \( 3234 a + 19113\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(-67a+79\right){x}+3234a+19113$ |
| 24.1-b2 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$7.270694035$ |
0.963026950 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( -7\bigr] \) |
${y}^2={x}^3+6{x}-7$ |
| 24.1-b3 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$7.270694035$ |
0.963026950 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 13 a + 44\) , \( -111 a - 470\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(13a+44\right){x}-111a-470$ |
| 24.1-b4 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$3.635347017$ |
0.963026950 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 93 a + 9\) , \( 324 a + 3915\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(93a+9\right){x}+324a+3915$ |
| 24.1-b5 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.635347017$ |
0.963026950 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 253 a - 61\) , \( -4476 a - 23267\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(253a-61\right){x}-4476a-23267$ |
| 24.1-b6 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$1.817673508$ |
0.963026950 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 1533 a - 621\) , \( 42174 a + 298557\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(1533a-621\right){x}+42174a+298557$ |
| 24.1-c1 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{16} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$16$ |
\( 2^{5} \) |
$1$ |
$1.817673508$ |
1.926053901 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 16\) , \( 180\bigr] \) |
${y}^2={x}^3+{x}^2+16{x}+180$ |
| 24.1-c2 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$7.270694035$ |
1.926053901 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^3+{x}^2+{x}$ |
| 24.1-c3 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$7.270694035$ |
1.926053901 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -4\) , \( -4\bigr] \) |
${y}^2={x}^3+{x}^2-4{x}-4$ |
| 24.1-c4 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$16$ |
\( 2^{4} \) |
$1$ |
$3.635347017$ |
1.926053901 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -24\) , \( 36\bigr] \) |
${y}^2={x}^3+{x}^2-24{x}+36$ |
| 24.1-c5 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$3.635347017$ |
1.926053901 |
\( \frac{28756228}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -64\) , \( -220\bigr] \) |
${y}^2={x}^3+{x}^2-64{x}-220$ |
| 24.1-c6 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{4} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$16$ |
\( 2^{3} \) |
$1$ |
$1.817673508$ |
1.926053901 |
\( \frac{3065617154}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -384\) , \( 2772\bigr] \) |
${y}^2={x}^3+{x}^2-384{x}+2772$ |
| 24.1-d1 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{16} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$2.191113404$ |
$1.817673508$ |
4.220202522 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 16\) , \( -180\bigr] \) |
${y}^2={x}^3-{x}^2+16{x}-180$ |
| 24.1-d2 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.382226809$ |
$7.270694035$ |
4.220202522 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^3-{x}^2+{x}$ |
| 24.1-d3 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$8.764453619$ |
$7.270694035$ |
4.220202522 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -4\) , \( 4\bigr] \) |
${y}^2={x}^3-{x}^2-4{x}+4$ |
| 24.1-d4 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$4.382226809$ |
$3.635347017$ |
4.220202522 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -24\) , \( -36\bigr] \) |
${y}^2={x}^3-{x}^2-24{x}-36$ |
| 24.1-d5 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$17.52890723$ |
$3.635347017$ |
4.220202522 |
\( \frac{28756228}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -64\) , \( 220\bigr] \) |
${y}^2={x}^3-{x}^2-64{x}+220$ |
| 24.1-d6 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{4} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$8.764453619$ |
$1.817673508$ |
4.220202522 |
\( \frac{3065617154}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -384\) , \( -2772\bigr] \) |
${y}^2={x}^3-{x}^2-384{x}-2772$ |
*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.
