| 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-a1 |
192.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
192.1 |
\( 2^{6} \cdot 3 \) |
\( 2^{16} \cdot 3 \) |
$0.57614$ |
$(-2a+1), (2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$3.635347017$ |
0.524717144 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 11 a - 6\) , \( 11 a - 1\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(11a-6\right){x}+11a-1$ |
| 768.1-a1 |
768.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3 \) |
$0.81478$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$3.635347017$ |
1.049434289 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 11 a - 6\) , \( -11 a + 1\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(11a-6\right){x}-11a+1$ |
| 2304.1-a1 |
2304.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{7} \) |
$1.07231$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.098868579$ |
1.211782339 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 16 a - 32\) , \( -44 a + 48\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(16a-32\right){x}-44a+48$ |
| 2304.1-b1 |
2304.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{7} \) |
$1.07231$ |
$(-2a+1), (2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.098868579$ |
1.211782339 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 16 a - 32\) , \( 44 a - 48\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(16a-32\right){x}+44a-48$ |
| 12288.1-b2 |
12288.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12288.1 |
\( 2^{12} \cdot 3 \) |
\( 2^{28} \cdot 3 \) |
$1.62956$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.079273864$ |
$1.817673508$ |
2.265254003 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -23 a - 20\) , \( -68 a - 35\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-23a-20\right){x}-68a-35$ |
| 12288.1-g2 |
12288.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12288.1 |
\( 2^{12} \cdot 3 \) |
\( 2^{28} \cdot 3 \) |
$1.62956$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.817673508$ |
2.098868579 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -20 a + 43\) , \( 68 a + 35\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-20a+43\right){x}+68a+35$ |
| 28224.1-c2 |
28224.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
28224.1 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{7} \cdot 7^{6} \) |
$2.00611$ |
$(-2a+1), (-3a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.220721263$ |
$0.793297756$ |
3.234966217 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -170 a + 211\) , \( -277 a - 975\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-170a+211\right){x}-277a-975$ |
| 28224.3-c2 |
28224.3-c |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
28224.3 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{7} \cdot 7^{6} \) |
$2.00611$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.220721263$ |
$0.793297756$ |
3.234966217 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -180 a - 24\) , \( -1320 a + 480\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-180a-24\right){x}-1320a+480$ |
| 32448.1-e2 |
32448.1-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32448.1 |
\( 2^{6} \cdot 3 \cdot 13^{2} \) |
\( 2^{16} \cdot 3 \cdot 13^{6} \) |
$2.07729$ |
$(-2a+1), (-4a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.008263852$ |
2.328485625 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -120 a\) , \( 548 a - 292\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-120a{x}+548a-292$ |
| 32448.3-e2 |
32448.3-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32448.3 |
\( 2^{6} \cdot 3 \cdot 13^{2} \) |
\( 2^{16} \cdot 3 \cdot 13^{6} \) |
$2.07729$ |
$(-2a+1), (4a-3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.008263852$ |
2.328485625 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -115 a + 125\) , \( 107 a + 470\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-115a+125\right){x}+107a+470$ |
| 36864.1-l2 |
36864.1-l |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
36864.1 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{7} \) |
$2.14462$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.393090629$ |
$1.049434289$ |
3.376245242 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -127 a + 68\) , \( 286 a - 445\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-127a+68\right){x}+286a-445$ |
| 36864.1-m2 |
36864.1-m |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
36864.1 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{7} \) |
$2.14462$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.348272657$ |
$1.049434289$ |
3.376245242 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -127 a + 68\) , \( -286 a + 445\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-127a+68\right){x}-286a+445$ |
| 37632.1-e2 |
37632.1-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.1 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{16} \cdot 3 \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (-3a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.374032019$ |
1.586595513 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 57 a - 70\) , \( 205 a - 142\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(57a-70\right){x}+205a-142$ |
| 37632.1-k2 |
37632.1-k |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.1 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{16} \cdot 3 \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (-3a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.572326543$ |
$1.374032019$ |
4.081241752 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 13 a + 57\) , \( -205 a + 142\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(13a+57\right){x}-205a+142$ |
| 37632.3-e2 |
37632.3-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.3 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{16} \cdot 3 \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.374032019$ |
1.586595513 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -68 a + 60\) , \( -40 a + 240\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-68a+60\right){x}-40a+240$ |
| 37632.3-k2 |
37632.3-k |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.3 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{16} \cdot 3 \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.643081635$ |
$1.374032019$ |
4.081241752 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -68 a + 60\) , \( 40 a - 240\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-68a+60\right){x}+40a-240$ |
| 112896.1-p2 |
112896.1-p |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
112896.1 |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{7} \cdot 7^{6} \) |
$2.83706$ |
$(-2a+1), (-3a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.365102045$ |
$0.793297756$ |
4.332967921 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -170 a + 211\) , \( 277 a + 975\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-170a+211\right){x}+277a+975$ |
| 112896.3-p2 |
112896.3-p |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
112896.3 |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{7} \cdot 7^{6} \) |
$2.83706$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.591275511$ |
$0.793297756$ |
4.332967921 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -180 a - 24\) , \( 1320 a - 480\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-180a-24\right){x}+1320a-480$ |
| 120000.1-h2 |
120000.1-h |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
120000.1 |
\( 2^{6} \cdot 3 \cdot 5^{4} \) |
\( 2^{16} \cdot 3 \cdot 5^{12} \) |
$2.88068$ |
$(-2a+1), (2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.331860695$ |
$0.727069403$ |
5.594510179 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -125 a + 267\) , \( 1125 a + 413\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-125a+267\right){x}+1125a+413$ |
| 129792.1-d2 |
129792.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
129792.1 |
\( 2^{8} \cdot 3 \cdot 13^{2} \) |
\( 2^{16} \cdot 3 \cdot 13^{6} \) |
$2.93773$ |
$(-2a+1), (-4a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.391541286$ |
$1.008263852$ |
3.948577566 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 120\) , \( -548 a + 292\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+120{x}-548a+292$ |
| 129792.3-d2 |
129792.3-d |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
129792.3 |
\( 2^{8} \cdot 3 \cdot 13^{2} \) |
\( 2^{16} \cdot 3 \cdot 13^{6} \) |
$2.93773$ |
$(-2a+1), (4a-3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.391541286$ |
$1.008263852$ |
3.948577566 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 125 a - 10\) , \( -107 a - 470\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(125a-10\right){x}-107a-470$ |
*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.
