| 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 |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{16} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$2.078568110$ |
$3.635347017$ |
2.661186243 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 16\) , \( -180\bigr] \) |
${y}^2={x}^3-{x}^2+16{x}-180$ |
| 24.1-a2 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.157136221$ |
$14.54138807$ |
2.661186243 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^3-{x}^2+{x}$ |
| 24.1-a3 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$8.314272442$ |
$14.54138807$ |
2.661186243 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -4\) , \( 4\bigr] \) |
${y}^2={x}^3-{x}^2-4{x}+4$ |
| 24.1-a4 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$4.157136221$ |
$7.270694035$ |
2.661186243 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -24\) , \( -36\bigr] \) |
${y}^2={x}^3-{x}^2-24{x}-36$ |
| 24.1-a5 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$16.62854488$ |
$7.270694035$ |
2.661186243 |
\( \frac{28756228}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -64\) , \( 220\bigr] \) |
${y}^2={x}^3-{x}^2-64{x}+220$ |
| 24.1-a6 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{4} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$8.314272442$ |
$3.635347017$ |
2.661186243 |
\( \frac{3065617154}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -384\) , \( -2772\bigr] \) |
${y}^2={x}^3-{x}^2-384{x}-2772$ |
| 24.1-b1 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$3.635347017$ |
0.640148915 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -167 a + 1371\) , \( 25242 a - 14271\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(-167a+1371\right){x}+25242a-14271$ |
| 24.1-b2 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$14.54138807$ |
0.640148915 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( -7\bigr] \) |
${y}^2={x}^3+6{x}-7$ |
| 24.1-b3 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$14.54138807$ |
0.640148915 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 33 a + 16\) , \( -791 a + 1194\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(33a+16\right){x}-791a+1194$ |
| 24.1-b4 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$7.270694035$ |
0.640148915 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 233 a - 1339\) , \( 3164 a + 9379\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(233a-1339\right){x}+3164a+9379$ |
| 24.1-b5 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$7.270694035$ |
0.640148915 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 633 a - 4049\) , \( -33908 a + 36669\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(633a-4049\right){x}-33908a+36669$ |
| 24.1-b6 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$3.635347017$ |
0.640148915 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 3833 a - 25729\) , \( 344246 a + 91189\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(3833a-25729\right){x}+344246a+91189$ |
| 24.1-c1 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{16} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$16$ |
\( 2^{5} \) |
$1$ |
$3.635347017$ |
1.280297830 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 16\) , \( 180\bigr] \) |
${y}^2={x}^3+{x}^2+16{x}+180$ |
| 24.1-c2 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$14.54138807$ |
1.280297830 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^3+{x}^2+{x}$ |
| 24.1-c3 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$14.54138807$ |
1.280297830 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -4\) , \( -4\bigr] \) |
${y}^2={x}^3+{x}^2-4{x}-4$ |
| 24.1-c4 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$16$ |
\( 2^{4} \) |
$1$ |
$7.270694035$ |
1.280297830 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -24\) , \( 36\bigr] \) |
${y}^2={x}^3+{x}^2-24{x}+36$ |
| 24.1-c5 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$7.270694035$ |
1.280297830 |
\( \frac{28756228}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -64\) , \( -220\bigr] \) |
${y}^2={x}^3+{x}^2-64{x}-220$ |
| 24.1-c6 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{4} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$16$ |
\( 2^{3} \) |
$1$ |
$3.635347017$ |
1.280297830 |
\( \frac{3065617154}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -384\) , \( 2772\bigr] \) |
${y}^2={x}^3+{x}^2-384{x}+2772$ |
| 24.1-d1 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
|
\( 2^{5} \) |
$1$ |
$3.635347017$ |
11.39152378 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -147 a + 1456\) , \( -21956 a - 10078\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-147a+1456\right){x}-21956a-10078$ |
| 24.1-d2 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
|
\( 2^{2} \) |
$1$ |
$14.54138807$ |
11.39152378 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( 7\bigr] \) |
${y}^2={x}^3+6{x}+7$ |
| 24.1-d3 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
|
\( 2^{3} \) |
$1$ |
$14.54138807$ |
11.39152378 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 53 a + 101\) , \( -123 a + 2912\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(53a+101\right){x}-123a+2912$ |
| 24.1-d4 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
|
\( 2^{5} \) |
$1$ |
$7.270694035$ |
11.39152378 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 253 a - 1254\) , \( -8278 a + 23182\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(253a-1254\right){x}-8278a+23182$ |
| 24.1-d5 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
|
\( 2^{3} \) |
$1$ |
$7.270694035$ |
11.39152378 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 653 a - 3964\) , \( 20394 a + 52802\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(653a-3964\right){x}+20394a+52802$ |
| 24.1-d6 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-129}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \cdot 23^{12} \) |
$4.49279$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
|
\( 2^{3} \) |
$1$ |
$3.635347017$ |
11.39152378 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 3853 a - 25644\) , \( -424960 a + 453562\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(3853a-25644\right){x}-424960a+453562$ |
*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.
