| 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 |
| 108.2-a1 |
108.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{13} \) |
$0.81478$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{4} \) |
$1$ |
$1.488316936$ |
1.052398998 |
\( -\frac{18202756}{81} a - \frac{253086988}{81} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -59 a + 4\) , \( 122 a - 261\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-59a+4\right){x}+122a-261$ |
| 108.2-a2 |
108.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{13} \) |
$0.81478$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$1.488316936$ |
1.052398998 |
\( \frac{18202756}{81} a - \frac{253086988}{81} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -48 a + 49\) , \( 7 a + 265\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-48a+49\right){x}+7a+265$ |
| 108.2-a3 |
108.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{10} \) |
$0.81478$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.976633872$ |
1.052398998 |
\( -\frac{372736}{27} a - \frac{352256}{27} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -4 a - 7\) , \( 11 a + 4\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a-7\right){x}+11a+4$ |
| 108.2-a4 |
108.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{10} \) |
$0.81478$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.976633872$ |
1.052398998 |
\( \frac{372736}{27} a - \frac{352256}{27} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 9\) , \( 5 a - 2\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+9{x}+5a-2$ |
| 108.2-a5 |
108.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{14} \) |
$0.81478$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$2.976633872$ |
1.052398998 |
\( -\frac{855712}{729} a + \frac{467888}{729} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -3 a + 4\) , \( -2 a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a+4\right){x}-2a+4$ |
| 108.2-a6 |
108.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{14} \) |
$0.81478$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$2.976633872$ |
1.052398998 |
\( \frac{855712}{729} a + \frac{467888}{729} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -4 a - 1\) , \( a - 7\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a-1\right){x}+a-7$ |
| 108.2-a7 |
108.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{19} \) |
$0.81478$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$1.488316936$ |
1.052398998 |
\( -\frac{715706108}{531441} a + \frac{421307996}{531441} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 11 a + 14\) , \( 16 a - 19\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(11a+14\right){x}+16a-19$ |
| 108.2-a8 |
108.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{19} \) |
$0.81478$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/12\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$1$ |
$1.488316936$ |
1.052398998 |
\( \frac{715706108}{531441} a + \frac{421307996}{531441} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 2 a - 21\) , \( 13 a + 37\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a-21\right){x}+13a+37$ |
*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.
