| 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-a4 |
192.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
192.1 |
\( 2^{6} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$0.57614$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$7.270694035$ |
0.524717144 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( a - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(a-1\right){x}$ |
| 768.1-a4 |
768.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$0.81478$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$7.270694035$ |
1.049434289 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( a - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(a-1\right){x}$ |
| 2304.1-a4 |
2304.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$1.07231$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$4.197737158$ |
1.211782339 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a - 2\) , \( -2 a\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a-2\right){x}-2a$ |
| 2304.1-b4 |
2304.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$1.07231$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$4.197737158$ |
1.211782339 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a - 2\) , \( 2 a\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a-2\right){x}+2a$ |
| 12288.1-b4 |
12288.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12288.1 |
\( 2^{12} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$1.62956$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.539636932$ |
$3.635347017$ |
2.265254003 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -3 a\) , \( -3\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}-3a{x}-3$ |
| 12288.1-g4 |
12288.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12288.1 |
\( 2^{12} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$1.62956$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.635347017$ |
2.098868579 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 3\) , \( 3\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+3{x}+3$ |
| 28224.1-c4 |
28224.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
28224.1 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 7^{6} \) |
$2.00611$ |
$(-2a+1), (-3a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.441442526$ |
$1.586595513$ |
3.234966217 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -5 a + 16\) , \( 14 a - 36\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a+16\right){x}+14a-36$ |
| 28224.3-c4 |
28224.3-c |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
28224.3 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 7^{6} \) |
$2.00611$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.441442526$ |
$1.586595513$ |
3.234966217 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -15 a + 6\) , \( -24 a - 6\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-15a+6\right){x}-24a-6$ |
| 32448.1-e4 |
32448.1-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32448.1 |
\( 2^{6} \cdot 3 \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 13^{6} \) |
$2.07729$ |
$(-2a+1), (-4a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.016527704$ |
2.328485625 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -10 a + 5\) , \( 6 a + 6\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-10a+5\right){x}+6a+6$ |
| 32448.3-e4 |
32448.3-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
32448.3 |
\( 2^{6} \cdot 3 \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 13^{6} \) |
$2.07729$ |
$(-2a+1), (4a-3), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.016527704$ |
2.328485625 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -5 a + 10\) , \( -6 a + 12\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a+10\right){x}-6a+12$ |
| 36864.1-l4 |
36864.1-l |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
36864.1 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{8} \) |
$2.14462$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.696545314$ |
$2.098868579$ |
3.376245242 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -7 a + 8\) , \( 10 a - 1\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-7a+8\right){x}+10a-1$ |
| 36864.1-m4 |
36864.1-m |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
36864.1 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{8} \) |
$2.14462$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.696545314$ |
$2.098868579$ |
3.376245242 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -7 a + 8\) , \( -10 a + 1\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-7a+8\right){x}-10a+1$ |
| 37632.1-e4 |
37632.1-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.1 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (-3a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$2.748064039$ |
1.586595513 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 2 a - 5\) , \( 4 a + 2\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(2a-5\right){x}+4a+2$ |
| 37632.1-k4 |
37632.1-k |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.1 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (-3a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.286163271$ |
$2.748064039$ |
4.081241752 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 3 a + 2\) , \( -4 a - 2\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(3a+2\right){x}-4a-2$ |
| 37632.3-e4 |
37632.3-e |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.3 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$2.748064039$ |
1.586595513 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -3 a + 5\) , \( -4 a + 6\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-3a+5\right){x}-4a+6$ |
| 37632.3-k4 |
37632.3-k |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
37632.3 |
\( 2^{8} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{6} \) |
$2.15570$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.286163271$ |
$2.748064039$ |
4.081241752 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -3 a + 5\) , \( 4 a - 6\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-3a+5\right){x}+4a-6$ |
| 112896.1-p4 |
112896.1-p |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
112896.1 |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 7^{6} \) |
$2.83706$ |
$(-2a+1), (-3a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.182551022$ |
$1.586595513$ |
4.332967921 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -5 a + 16\) , \( -14 a + 36\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-5a+16\right){x}-14a+36$ |
| 112896.3-p4 |
112896.3-p |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
112896.3 |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 7^{6} \) |
$2.83706$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.182551022$ |
$1.586595513$ |
4.332967921 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -15 a + 6\) , \( 24 a + 6\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-15a+6\right){x}+24a+6$ |
| 120000.1-h4 |
120000.1-h |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
120000.1 |
\( 2^{6} \cdot 3 \cdot 5^{4} \) |
\( 2^{8} \cdot 3^{2} \cdot 5^{12} \) |
$2.88068$ |
$(-2a+1), (2), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.665930347$ |
$1.454138807$ |
5.594510179 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 17\) , \( 38\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+17{x}+38$ |
| 129792.1-d4 |
129792.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
129792.1 |
\( 2^{8} \cdot 3 \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 13^{6} \) |
$2.93773$ |
$(-2a+1), (-4a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.695770643$ |
$2.016527704$ |
3.948577566 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 5 a + 5\) , \( -6 a - 6\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(5a+5\right){x}-6a-6$ |
| 129792.3-d4 |
129792.3-d |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
129792.3 |
\( 2^{8} \cdot 3 \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 13^{6} \) |
$2.93773$ |
$(-2a+1), (4a-3), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.695770643$ |
$2.016527704$ |
3.948577566 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 10 a - 5\) , \( 6 a - 12\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(10a-5\right){x}+6a-12$ |
*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.
