| 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 |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$12.14060112$ |
$1.817673508$ |
5.313265514 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -73 a + 201\) , \( -4662 a - 25041\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a+1\right){x}^2+\left(-73a+201\right){x}-4662a-25041$ |
| 24.1-a2 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.517575141$ |
$7.270694035$ |
5.313265514 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( 7\bigr] \) |
${y}^2={x}^3+6{x}+7$ |
| 24.1-a3 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$3.035150282$ |
$7.270694035$ |
5.313265514 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 27 a + 46\) , \( -49 a + 654\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a+1\right){x}^2+\left(27a+46\right){x}-49a+654$ |
| 24.1-a4 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$6.070300564$ |
$3.635347017$ |
5.313265514 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 127 a - 109\) , \( -1904 a - 3611\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a+1\right){x}^2+\left(127a-109\right){x}-1904a-3611$ |
| 24.1-a5 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$6.070300564$ |
$3.635347017$ |
5.313265514 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 327 a - 419\) , \( 4088 a + 32799\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a+1\right){x}^2+\left(327a-419\right){x}+4088a+32799$ |
| 24.1-a6 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$3.035150282$ |
$1.817673508$ |
5.313265514 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 1927 a - 2899\) , \( -93506 a - 350021\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a+1\right){x}^2+\left(1927a-2899\right){x}-93506a-350021$ |
| 24.1-b1 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$2.170705473$ |
$1.817673508$ |
7.599975922 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -85 a + 246\) , \( 5525 a + 23452\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-85a+246\right){x}+5525a+23452$ |
| 24.1-b2 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$17.36564378$ |
$7.270694035$ |
7.599975922 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( -7\bigr] \) |
${y}^2={x}^3+6{x}-7$ |
| 24.1-b3 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$8.682821893$ |
$7.270694035$ |
7.599975922 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 15 a + 91\) , \( -188 a - 538\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(15a+91\right){x}-188a-538$ |
| 24.1-b4 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$4.341410946$ |
$3.635347017$ |
7.599975922 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 115 a - 64\) , \( 567 a + 5432\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(115a-64\right){x}+567a+5432$ |
| 24.1-b5 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.341410946$ |
$3.635347017$ |
7.599975922 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 315 a - 374\) , \( -7625 a - 27568\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(315a-374\right){x}-7625a-27568$ |
| 24.1-b6 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \cdot 13^{12} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$2.170705473$ |
$1.817673508$ |
7.599975922 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 1915 a - 2854\) , \( 72369 a + 382532\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(1915a-2854\right){x}+72369a+382532$ |
| 24.1-c1 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{16} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$20.16017604$ |
$1.817673508$ |
4.411493590 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 16\) , \( 180\bigr] \) |
${y}^2={x}^3+{x}^2+16{x}+180$ |
| 24.1-c2 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.520022005$ |
$7.270694035$ |
4.411493590 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^3+{x}^2+{x}$ |
| 24.1-c3 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$5.040044010$ |
$7.270694035$ |
4.411493590 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -4\) , \( -4\bigr] \) |
${y}^2={x}^3+{x}^2-4{x}-4$ |
| 24.1-c4 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$10.08008802$ |
$3.635347017$ |
4.411493590 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -24\) , \( 36\bigr] \) |
${y}^2={x}^3+{x}^2-24{x}+36$ |
| 24.1-c5 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.520022005$ |
$3.635347017$ |
4.411493590 |
\( \frac{28756228}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -64\) , \( -220\bigr] \) |
${y}^2={x}^3+{x}^2-64{x}-220$ |
| 24.1-c6 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{4} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$5.040044010$ |
$1.817673508$ |
4.411493590 |
\( \frac{3065617154}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -384\) , \( 2772\bigr] \) |
${y}^2={x}^3+{x}^2-384{x}+2772$ |
| 24.1-d1 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{16} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$4.499274317$ |
$1.817673508$ |
1.969081993 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 16\) , \( -180\bigr] \) |
${y}^2={x}^3-{x}^2+16{x}-180$ |
| 24.1-d2 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.249637158$ |
$7.270694035$ |
1.969081993 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^3-{x}^2+{x}$ |
| 24.1-d3 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$4.499274317$ |
$7.270694035$ |
1.969081993 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -4\) , \( 4\bigr] \) |
${y}^2={x}^3-{x}^2-4{x}+4$ |
| 24.1-d4 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$2.249637158$ |
$3.635347017$ |
1.969081993 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -24\) , \( -36\bigr] \) |
${y}^2={x}^3-{x}^2-24{x}-36$ |
| 24.1-d5 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$2.249637158$ |
$3.635347017$ |
1.969081993 |
\( \frac{28756228}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -64\) , \( 220\bigr] \) |
${y}^2={x}^3-{x}^2-64{x}+220$ |
| 24.1-d6 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{4} \) |
$3.28584$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1.124818579$ |
$1.817673508$ |
1.969081993 |
\( \frac{3065617154}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -384\) , \( -2772\bigr] \) |
${y}^2={x}^3-{x}^2-384{x}-2772$ |
| 36.1-a1 |
36.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{18} \) |
$3.63638$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$3.908998117$ |
$5.108115717$ |
7.211454998 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 27\bigr] \) |
${y}^2={x}^3+27$ |
| 36.1-a2 |
36.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{6} \) |
$3.63638$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$11.72699435$ |
$5.108115717$ |
7.211454998 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -1\bigr] \) |
${y}^2={x}^3-1$ |
| 36.1-a3 |
36.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{6} \) |
$3.63638$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$5.863497176$ |
$5.108115717$ |
7.211454998 |
\( 54000 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -15\) , \( -22\bigr] \) |
${y}^2={x}^3-15{x}-22$ |
| 36.1-a4 |
36.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{6} \cdot 13^{12} \) |
$3.63638$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.954499058$ |
$5.108115717$ |
7.211454998 |
\( 54000 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 56 a - 26\) , \( 172 a + 1164\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(56a-26\right){x}+172a+1164$ |
| 36.1-b1 |
36.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{18} \) |
$3.63638$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
|
\( 2 \) |
$1$ |
$5.108115717$ |
4.017703663 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -27\bigr] \) |
${y}^2={x}^3-27$ |
| 36.1-b2 |
36.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{6} \) |
$3.63638$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
|
\( 2 \cdot 3 \) |
$1$ |
$5.108115717$ |
4.017703663 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 1\bigr] \) |
${y}^2={x}^3+1$ |
| 36.1-b3 |
36.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{6} \) |
$3.63638$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
|
\( 2^{2} \cdot 3 \) |
$1$ |
$5.108115717$ |
4.017703663 |
\( 54000 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -15\) , \( 22\bigr] \) |
${y}^2={x}^3-15{x}+22$ |
| 36.1-b4 |
36.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{6} \cdot 13^{12} \) |
$3.63638$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
|
\( 2^{2} \) |
$1$ |
$5.108115717$ |
4.017703663 |
\( 54000 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 57 a - 60\) , \( -1098 a - 4738\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(57a-60\right){x}-1098a-4738$ |
| 46.1-a1 |
46.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
46.1 |
\( 2 \cdot 23 \) |
\( 2^{20} \cdot 23^{2} \) |
$3.86618$ |
$(2,a+1), (23,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$2.403591625$ |
11.57433713 |
\( -\frac{116930169}{23552} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 89\) , \( -117\bigr] \) |
${y}^2+a{x}{y}={x}^3+89{x}-117$ |
| 46.1-a2 |
46.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
46.1 |
\( 2 \cdot 23 \) |
\( 2^{10} \cdot 23^{4} \) |
$3.86618$ |
$(2,a+1), (23,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$1.201795812$ |
11.57433713 |
\( \frac{545138290809}{16928} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -71\) , \( 1643\bigr] \) |
${y}^2+a{x}{y}={x}^3-71{x}+1643$ |
| 46.1-b1 |
46.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
46.1 |
\( 2 \cdot 23 \) |
\( 2^{20} \cdot 23^{2} \) |
$3.86618$ |
$(2,a+1), (23,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$2.403591625$ |
1.157433713 |
\( -\frac{116930169}{23552} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -10\) , \( -12\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-10{x}-12$ |
| 46.1-b2 |
46.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
46.1 |
\( 2 \cdot 23 \) |
\( 2^{10} \cdot 23^{4} \) |
$3.86618$ |
$(2,a+1), (23,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$1.201795812$ |
1.157433713 |
\( \frac{545138290809}{16928} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -170\) , \( -812\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-170{x}-812$ |
| 46.1-c1 |
46.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
46.1 |
\( 2 \cdot 23 \) |
\( 2^{20} \cdot 3^{12} \cdot 23^{2} \) |
$3.86618$ |
$(2,a+1), (23,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$2.403591625$ |
1.157433713 |
\( -\frac{116930169}{23552} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 42\) , \( -38\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+42{x}-38$ |
| 46.1-c2 |
46.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
46.1 |
\( 2 \cdot 23 \) |
\( 2^{10} \cdot 3^{12} \cdot 23^{4} \) |
$3.86618$ |
$(2,a+1), (23,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$1.201795812$ |
1.157433713 |
\( \frac{545138290809}{16928} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -1398\) , \( -14438\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-1398{x}-14438$ |
| 46.1-d1 |
46.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
46.1 |
\( 2 \cdot 23 \) |
\( 2^{20} \cdot 3^{12} \cdot 23^{2} \) |
$3.86618$ |
$(2,a+1), (23,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.406132117$ |
$2.403591625$ |
4.700710045 |
\( -\frac{116930169}{23552} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -92\) , \( 415\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-92{x}+415$ |
| 46.1-d2 |
46.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
46.1 |
\( 2 \cdot 23 \) |
\( 2^{10} \cdot 3^{12} \cdot 23^{4} \) |
$3.86618$ |
$(2,a+1), (23,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1.624528469$ |
$1.201795812$ |
4.700710045 |
\( \frac{545138290809}{16928} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1532\) , \( 23455\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-1532{x}+23455$ |
| 64.1-a1 |
64.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \cdot 3^{12} \) |
$4.19892$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$10.36787921$ |
$6.875185818$ |
8.581235565 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -9\) , \( 0\bigr] \) |
${y}^2={x}^3-9{x}$ |
| 64.1-a2 |
64.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \cdot 13^{12} \) |
$4.19892$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$5.183939605$ |
$6.875185818$ |
8.581235565 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -20 a + 31\) , \( 0\bigr] \) |
${y}^2={x}^3+\left(-20a+31\right){x}$ |
| 64.1-a3 |
64.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{6} \cdot 13^{12} \) |
$4.19892$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 1 \) |
$5.183939605$ |
$6.875185818$ |
8.581235565 |
\( 287496 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 37 a - 29\) , \( 77 a + 581\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(37a-29\right){x}+77a+581$ |
| 64.1-a4 |
64.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{6} \cdot 13^{12} \) |
$4.19892$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 1 \) |
$5.183939605$ |
$6.875185818$ |
8.581235565 |
\( 287496 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 36 a + 5\) , \( -732 a - 3147\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(36a+5\right){x}-732a-3147$ |
| 64.1-b1 |
64.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \) |
$4.19892$ |
$(2,a+1)$ |
$0 \le r \le 1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
|
\( 2 \) |
$1$ |
$6.875185818$ |
4.610782267 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^3-{x}$ |
| 64.1-b2 |
64.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{24} \) |
$4.19892$ |
$(2,a+1)$ |
$0 \le r \le 1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
|
\( 2^{2} \) |
$1$ |
$6.875185818$ |
4.610782267 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4\) , \( 0\bigr] \) |
${y}^2={x}^3+4{x}$ |
| 64.1-b3 |
64.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{18} \) |
$4.19892$ |
$(2,a+1)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
|
\( 1 \) |
$1$ |
$6.875185818$ |
4.610782267 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -11\) , \( -14\bigr] \) |
${y}^2={x}^3-11{x}-14$ |
| 64.1-b4 |
64.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{18} \) |
$4.19892$ |
$(2,a+1)$ |
$0 \le r \le 1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
|
\( 1 \) |
$1$ |
$6.875185818$ |
4.610782267 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -11\) , \( 14\bigr] \) |
${y}^2={x}^3-11{x}+14$ |
| 69.1-a1 |
69.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3^{16} \cdot 23^{2} \) |
$4.27864$ |
$(3,a), (23,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.555095912$ |
$5.742727531$ |
14.13158451 |
\( -\frac{15625}{207} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 129\) , \( -165\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+129{x}-165$ |
| 69.1-a2 |
69.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-69}) \) |
$2$ |
$[0, 1]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3^{14} \cdot 23^{4} \) |
$4.27864$ |
$(3,a), (23,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$10.22038364$ |
$2.871363765$ |
14.13158451 |
\( \frac{413493625}{1587} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -6\) , \( -3\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-6{x}-3$ |
*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.
