| 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 |
| 576.2-a5 |
576.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
576.2 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$1.23820$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.846792627$ |
1.360046576 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 2 a + 1\) , \( a - 4\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(2a+1\right){x}+a-4$ |
| 576.2-b5 |
576.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
576.2 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$1.23820$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$3.846792627$ |
1.360046576 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 2 a + 1\) , \( -a + 4\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(2a+1\right){x}-a+4$ |
| 5184.3-a5 |
5184.3-a |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
5184.3 |
\( 2^{6} \cdot 3^{4} \) |
\( 2^{12} \cdot 3^{18} \) |
$2.14462$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.471543832$ |
$1.282264209$ |
3.420381731 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 18 a + 15\) , \( 14 a - 72\bigr] \) |
${y}^2={x}^{3}+\left(18a+15\right){x}+14a-72$ |
| 5184.3-b5 |
5184.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
5184.3 |
\( 2^{6} \cdot 3^{4} \) |
\( 2^{12} \cdot 3^{18} \) |
$2.14462$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.943087664$ |
$1.282264209$ |
3.420381731 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 18 a + 15\) , \( -14 a + 72\bigr] \) |
${y}^2={x}^{3}+\left(18a+15\right){x}-14a+72$ |
| 6912.2-b5 |
6912.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.2 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{12} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.220946758$ |
1.570446513 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 2 a - 10\) , \( -8 a + 8\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(2a-10\right){x}-8a+8$ |
| 6912.2-m5 |
6912.2-m |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.2 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{12} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.383567238$ |
$2.220946758$ |
4.345636691 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 2 a - 10\) , \( 8 a - 8\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(2a-10\right){x}+8a-8$ |
| 6912.3-e5 |
6912.3-e |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.3 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{12} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.220946758$ |
1.570446513 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -6 a + 6\) , \( -4 a - 16\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6a+6\right){x}-4a-16$ |
| 6912.3-j5 |
6912.3-j |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.3 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{12} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.691783619$ |
$2.220946758$ |
4.345636691 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -6 a + 6\) , \( 4 a + 16\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-6a+6\right){x}+4a+16$ |
| 9216.2-a5 |
9216.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{6} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.504669137$ |
$2.720093152$ |
3.882715037 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -4 a - 3\) , \( -4 a - 1\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-4a-3\right){x}-4a-1$ |
| 9216.2-bb5 |
9216.2-bb |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{6} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$2.720093152$ |
3.846792627 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -4 a - 3\) , \( 4 a + 1\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-4a-3\right){x}+4a+1$ |
| 27648.2-w5 |
27648.2-w |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{12} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.570446513$ |
2.220946758 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -2 a + 19\) , \( -19 a - 13\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-2a+19\right){x}-19a-13$ |
| 27648.2-z5 |
27648.2-z |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{12} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.173509605$ |
$1.570446513$ |
5.212604708 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2 a + 19\) , \( 19 a + 13\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a+19\right){x}+19a+13$ |
| 27648.3-w5 |
27648.3-w |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{12} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.570446513$ |
2.220946758 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 10 a - 13\) , \( -21 a + 3\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(10a-13\right){x}-21a+3$ |
| 27648.3-z5 |
27648.3-z |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{12} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.586754802$ |
$1.570446513$ |
5.212604708 |
\( -\frac{121088}{81} a - \frac{65728}{81} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 10 a - 13\) , \( 21 a - 3\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(10a-13\right){x}+21a-3$ |
*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.
