| 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 |
| 3456.3-a1 |
3456.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{7} \cdot 3^{21} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.186367624$ |
0.838888592 |
\( -\frac{5577343220}{531441} a - \frac{1362907864}{531441} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 6 a - 54\) , \( 47 a - 136\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(6a-54\right){x}+47a-136$ |
| 3456.3-a2 |
3456.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{7} \cdot 3^{21} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.186367624$ |
0.838888592 |
\( \frac{5577343220}{531441} a - \frac{1362907864}{531441} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -19 a + 46\) , \( -93 a - 125\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-19a+46\right){x}-93a-125$ |
| 3456.3-a3 |
3456.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{14} \cdot 3^{18} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.186367624$ |
0.838888592 |
\( -\frac{219488}{729} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 12 a + 6\) , \( 18 a + 90\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(12a+6\right){x}+18a+90$ |
| 3456.3-a4 |
3456.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{16} \cdot 3^{12} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.186367624$ |
0.838888592 |
\( \frac{19056256}{27} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 70 a + 35\) , \( -69 a - 345\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(70a+35\right){x}-69a-345$ |
| 3456.3-b1 |
3456.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{16} \cdot 3^{11} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.267942897$ |
$1.698770867$ |
2.574850644 |
\( -\frac{167792}{9} a - \frac{21616}{9} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -8 a - 28\) , \( 28 a + 40\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a-28\right){x}+28a+40$ |
| 3456.3-b2 |
3456.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{10} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.535885794$ |
$3.397541734$ |
2.574850644 |
\( \frac{3584}{3} a + \frac{4096}{3} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -3 a - 3\) , \( -2 a - 2\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a-3\right){x}-2a-2$ |
| 3456.3-c1 |
3456.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{16} \cdot 3^{9} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.201672578$ |
$2.398280568$ |
2.736036130 |
\( \frac{335248}{729} a + \frac{350000}{729} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -4 a + 4\) , \( -8 a - 4\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a+4\right){x}-8a-4$ |
| 3456.3-c2 |
3456.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{6} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.403345156$ |
$4.796561136$ |
2.736036130 |
\( -\frac{48640}{27} a + \frac{74752}{27} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( a - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(a-1\right){x}$ |
| 3456.3-d1 |
3456.3-d |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{7} \cdot 3^{11} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.478326877$ |
1.752441741 |
\( -\frac{4137500}{81} a - \frac{12439000}{81} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -13 a + 5\) , \( -9 a + 21\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-13a+5\right){x}-9a+21$ |
| 3456.3-d2 |
3456.3-d |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{7} \cdot 3^{11} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.478326877$ |
1.752441741 |
\( \frac{4137500}{81} a - \frac{12439000}{81} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -8 a - 15\) , \( 26 a + 16\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-8a-15\right){x}+26a+16$ |
| 3456.3-d3 |
3456.3-d |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{14} \cdot 3^{10} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.478326877$ |
1.752441741 |
\( \frac{4000}{9} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -4 a - 2\) , \( -2 a - 10\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a-2\right){x}-2a-10$ |
| 3456.3-d4 |
3456.3-d |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{16} \cdot 3^{8} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.478326877$ |
1.752441741 |
\( \frac{16000}{3} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 6 a + 3\) , \( 3 a + 15\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(6a+3\right){x}+3a+15$ |
| 3456.3-e1 |
3456.3-e |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{7} \cdot 3^{11} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.478326877$ |
1.752441741 |
\( -\frac{4137500}{81} a - \frac{12439000}{81} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -13 a + 5\) , \( 9 a - 20\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-13a+5\right){x}+9a-20$ |
| 3456.3-e2 |
3456.3-e |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{7} \cdot 3^{11} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.478326877$ |
1.752441741 |
\( \frac{4137500}{81} a - \frac{12439000}{81} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -8 a - 15\) , \( -26 a - 15\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-8a-15\right){x}-26a-15$ |
| 3456.3-e3 |
3456.3-e |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{14} \cdot 3^{10} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.478326877$ |
1.752441741 |
\( \frac{4000}{9} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -4 a - 2\) , \( 2 a + 10\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a-2\right){x}+2a+10$ |
| 3456.3-e4 |
3456.3-e |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{16} \cdot 3^{8} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.478326877$ |
1.752441741 |
\( \frac{16000}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 6 a + 3\) , \( -3 a - 15\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(6a+3\right){x}-3a-15$ |
| 3456.3-f1 |
3456.3-f |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{16} \cdot 3^{9} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.086346921$ |
$2.398280568$ |
3.514334458 |
\( \frac{335248}{729} a + \frac{350000}{729} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -4 a + 4\) , \( 8 a + 4\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a+4\right){x}+8a+4$ |
| 3456.3-f2 |
3456.3-f |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{6} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.172693842$ |
$4.796561136$ |
3.514334458 |
\( -\frac{48640}{27} a + \frac{74752}{27} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( a - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(a-1\right){x}$ |
| 3456.3-g1 |
3456.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{16} \cdot 3^{11} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.698770867$ |
2.402424800 |
\( -\frac{167792}{9} a - \frac{21616}{9} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -8 a - 28\) , \( -28 a - 40\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-8a-28\right){x}-28a-40$ |
| 3456.3-g2 |
3456.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{10} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.397541734$ |
2.402424800 |
\( \frac{3584}{3} a + \frac{4096}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -3 a - 3\) , \( 2 a + 2\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-3a-3\right){x}+2a+2$ |
| 3456.3-h1 |
3456.3-h |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{7} \cdot 3^{21} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$1.186367624$ |
2.516665776 |
\( -\frac{5577343220}{531441} a - \frac{1362907864}{531441} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 6 a - 54\) , \( -47 a + 136\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a-54\right){x}-47a+136$ |
| 3456.3-h2 |
3456.3-h |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{7} \cdot 3^{21} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.186367624$ |
2.516665776 |
\( \frac{5577343220}{531441} a - \frac{1362907864}{531441} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -19 a + 46\) , \( 93 a + 125\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-19a+46\right){x}+93a+125$ |
| 3456.3-h3 |
3456.3-h |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{14} \cdot 3^{18} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$1.186367624$ |
2.516665776 |
\( -\frac{219488}{729} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 12 a + 6\) , \( -18 a - 90\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(12a+6\right){x}-18a-90$ |
| 3456.3-h4 |
3456.3-h |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3456.3 |
\( 2^{7} \cdot 3^{3} \) |
\( 2^{16} \cdot 3^{12} \) |
$1.93788$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.186367624$ |
2.516665776 |
\( \frac{19056256}{27} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 70 a + 35\) , \( 69 a + 345\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(70a+35\right){x}+69a+345$ |
*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.
