| 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 |
| 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 |
\( -\frac{73696}{3} a - \frac{550672}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 11 a - 6\) , \( -11 a + 1\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(11a-6\right){x}-11a+1$ |
| 768.1-a2 |
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 |
\( \frac{73696}{3} a - \frac{624368}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 6 a - 11\) , \( 11 a - 10\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(6a-11\right){x}+11a-10$ |
| 768.1-a3 |
768.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{22} \cdot 3^{16} \) |
$0.81478$ |
$(-2a+1), (2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.908836754$ |
1.049434289 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 16 a - 16\) , \( 180\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(16a-16\right){x}+180$ |
| 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 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( a - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(a-1\right){x}$ |
| 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 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -4 a + 4\) , \( -4\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-4a+4\right){x}-4$ |
| 768.1-a6 |
768.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$0.81478$ |
$(-2a+1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.817673508$ |
1.049434289 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -24 a + 24\) , \( 36\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-24a+24\right){x}+36$ |
| 768.1-a7 |
768.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$0.81478$ |
$(-2a+1), (2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.817673508$ |
1.049434289 |
\( \frac{28756228}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -64 a + 64\) , \( -220\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-64a+64\right){x}-220$ |
| 768.1-a8 |
768.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{22} \cdot 3^{4} \) |
$0.81478$ |
$(-2a+1), (2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.908836754$ |
1.049434289 |
\( \frac{3065617154}{9} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -384 a + 384\) , \( 2772\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-384a+384\right){x}+2772$ |
*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.
