| 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 |
| 192.1-a5 |
192.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
192.1 |
\( 2^{6} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$0.57614$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.635347017$ |
0.524717144 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -4 a + 4\) , \( 4\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-4a+4\right){x}+4$ |
| 768.1-a5 |
768.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$0.81478$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$3.635347017$ |
1.049434289 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -4 a + 4\) , \( -4\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-4a+4\right){x}-4$ |
| 2304.1-a5 |
2304.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{10} \) |
$1.07231$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$2.098868579$ |
1.211782339 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 13\) , \( -11 a + 12\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+13\right){x}-11a+12$ |
| 2304.1-b5 |
2304.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{10} \) |
$1.07231$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$2.098868579$ |
1.211782339 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 13\) , \( 11 a - 12\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+13\right){x}+11a-12$ |
| 12288.1-b5 |
12288.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12288.1 |
\( 2^{12} \cdot 3 \) |
\( 2^{28} \cdot 3^{4} \) |
$1.62956$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.079273864$ |
$1.817673508$ |
2.265254003 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 17 a\) , \( -15\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+17a{x}-15$ |
| 12288.1-g5 |
12288.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12288.1 |
\( 2^{12} \cdot 3 \) |
\( 2^{28} \cdot 3^{4} \) |
$1.62956$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.817673508$ |
2.098868579 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -17\) , \( 15\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-17{x}+15$ |
| 28224.1-c5 |
28224.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
28224.1 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{10} \cdot 7^{6} \) |
$2.00611$ |
$(-2a+1), (-3a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.882885052$ |
$0.793297756$ |
3.234966217 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 40 a - 104\) , \( 164 a - 240\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(40a-104\right){x}+164a-240$ |
| 28224.3-c5 |
28224.3-c |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
28224.3 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{10} \cdot 7^{6} \) |
$2.00611$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.882885052$ |
$0.793297756$ |
3.234966217 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 105 a - 39\) , \( -99 a - 180\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(105a-39\right){x}-99a-180$ |
| 32448.1-e5 |
32448.1-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32448.1 |
\( 2^{6} \cdot 3 \cdot 13^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 13^{6} \) |
$2.07729$ |
$(-2a+1), (-4a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.008263852$ |
2.328485625 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 65 a - 30\) , \( 115 a + 34\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(65a-30\right){x}+115a+34$ |
| 32448.3-e5 |
32448.3-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32448.3 |
\( 2^{6} \cdot 3 \cdot 13^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 13^{6} \) |
$2.07729$ |
$(-2a+1), (4a-3), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.008263852$ |
2.328485625 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 30 a - 65\) , \( -115 a + 149\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(30a-65\right){x}-115a+149$ |
| 36864.1-l5 |
36864.1-l |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
36864.1 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{10} \) |
$2.14462$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.393090629$ |
$1.049434289$ |
3.376245242 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 53 a - 52\) , \( 142 a - 97\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(53a-52\right){x}+142a-97$ |
| 36864.1-m5 |
36864.1-m |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
36864.1 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{10} \) |
$2.14462$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.393090629$ |
$1.049434289$ |
3.376245242 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 53 a - 52\) , \( -142 a + 97\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(53a-52\right){x}-142a+97$ |
| 37632.1-e5 |
37632.1-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.1 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (-3a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.374032019$ |
1.586595513 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -13 a + 35\) , \( 51 a - 9\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-13a+35\right){x}+51a-9$ |
| 37632.1-k5 |
37632.1-k |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.1 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (-3a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.643081635$ |
$1.374032019$ |
4.081241752 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -22 a - 13\) , \( -51 a + 9\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-22a-13\right){x}-51a+9$ |
| 37632.3-e5 |
37632.3-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.3 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.374032019$ |
1.586595513 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 22 a - 35\) , \( -51 a + 42\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(22a-35\right){x}-51a+42$ |
| 37632.3-k5 |
37632.3-k |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.3 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.643081635$ |
$1.374032019$ |
4.081241752 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 22 a - 35\) , \( 51 a - 42\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(22a-35\right){x}+51a-42$ |
| 112896.1-p5 |
112896.1-p |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
112896.1 |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{10} \cdot 7^{6} \) |
$2.83706$ |
$(-2a+1), (-3a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.591275511$ |
$0.793297756$ |
4.332967921 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 40 a - 104\) , \( -164 a + 240\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(40a-104\right){x}-164a+240$ |
| 112896.3-p5 |
112896.3-p |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
112896.3 |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{10} \cdot 7^{6} \) |
$2.83706$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.591275511$ |
$0.793297756$ |
4.332967921 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 105 a - 39\) , \( 99 a + 180\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(105a-39\right){x}+99a+180$ |
| 120000.1-h5 |
120000.1-h |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
120000.1 |
\( 2^{6} \cdot 3 \cdot 5^{4} \) |
\( 2^{16} \cdot 3^{4} \cdot 5^{12} \) |
$2.88068$ |
$(-2a+1), (2), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.832965173$ |
$0.727069403$ |
5.594510179 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -108\) , \( 288\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-108{x}+288$ |
| 129792.1-d5 |
129792.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
129792.1 |
\( 2^{8} \cdot 3 \cdot 13^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 13^{6} \) |
$2.93773$ |
$(-2a+1), (-4a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.847885321$ |
$1.008263852$ |
3.948577566 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -30 a - 35\) , \( -115 a - 34\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-30a-35\right){x}-115a-34$ |
| 129792.3-d5 |
129792.3-d |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
129792.3 |
\( 2^{8} \cdot 3 \cdot 13^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 13^{6} \) |
$2.93773$ |
$(-2a+1), (4a-3), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.847885321$ |
$1.008263852$ |
3.948577566 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -65 a + 35\) , \( 115 a - 149\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-65a+35\right){x}+115a-149$ |
*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.
