| 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 |
| 288.2-b3 |
288.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{8} \) |
$1.04119$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$3.353492831$ |
1.185638761 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 1\) , \( 3 a\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+{x}+3a$ |
| 288.2-c3 |
288.2-c |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{8} \) |
$1.04119$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$3.353492831$ |
1.185638761 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( -3 a\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+{x}-3a$ |
| 2592.3-b3 |
2592.3-b |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
2592.3 |
\( 2^{5} \cdot 3^{4} \) |
\( 2^{12} \cdot 3^{20} \) |
$1.80340$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.117830943$ |
1.580851681 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 15\) , \( -68 a\bigr] \) |
${y}^2={x}^{3}+15{x}-68a$ |
| 2592.3-f3 |
2592.3-f |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
2592.3 |
\( 2^{5} \cdot 3^{4} \) |
\( 2^{12} \cdot 3^{20} \) |
$1.80340$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.117830943$ |
1.580851681 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 15\) , \( 68 a\bigr] \) |
${y}^2={x}^{3}+15{x}+68a$ |
| 6912.2-e3 |
6912.2-e |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.2 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{14} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.936139989$ |
1.369057715 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4 a - 2\) , \( -12 a - 8\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(4a-2\right){x}-12a-8$ |
| 6912.2-j3 |
6912.2-j |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.2 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{14} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.788156716$ |
$1.936139989$ |
4.316128133 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 4 a - 2\) , \( 12 a + 8\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-2\right){x}+12a+8$ |
| 6912.3-b3 |
6912.3-b |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.3 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{14} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.936139989$ |
1.369057715 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -4 a - 2\) , \( 12 a - 8\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a-2\right){x}+12a-8$ |
| 6912.3-m3 |
6912.3-m |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.3 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{12} \cdot 3^{14} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.788156716$ |
$1.936139989$ |
4.316128133 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -4 a - 2\) , \( -12 a + 8\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a-2\right){x}-12a+8$ |
| 9216.2-h3 |
9216.2-h |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{8} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.878591722$ |
$2.371277522$ |
2.946351043 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -3\) , \( -9\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-3{x}-9$ |
| 9216.2-u3 |
9216.2-u |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{8} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.385972529$ |
$2.371277522$ |
5.177424439 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -3\) , \( 9\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-3{x}+9$ |
| 27648.2-c3 |
27648.2-c |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{14} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.880355775$ |
$1.369057715$ |
3.408984042 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -6 a + 3\) , \( 9 a - 45\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-6a+3\right){x}+9a-45$ |
| 27648.2-bu3 |
27648.2-bu |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{14} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$1.369057715$ |
3.872279978 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -6 a + 3\) , \( -9 a + 45\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a+3\right){x}-9a+45$ |
| 27648.3-b3 |
27648.3-b |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{14} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.880355775$ |
$1.369057715$ |
3.408984042 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 6 a + 3\) , \( -9 a - 45\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(6a+3\right){x}-9a-45$ |
| 27648.3-bt3 |
27648.3-bt |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{14} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$1.369057715$ |
3.872279978 |
\( -\frac{8000}{81} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 6 a + 3\) , \( 9 a + 45\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(6a+3\right){x}+9a+45$ |
*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.
