| 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 |
| 1.1-a1 |
1.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.65736$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$43$ |
43Ns.9.1 |
$1$ |
\( 1 \) |
$1$ |
$50.75994773$ |
0.684198241 |
\( 8000 \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -483 a - 4181\) , \( 12715 a + 119085\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-483a-4181\right){x}+12715a+119085$ |
| 1.1-a2 |
1.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.65736$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$43$ |
43Ns.9.1 |
$1$ |
\( 1 \) |
$1$ |
$50.75994773$ |
0.684198241 |
\( 8000 \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( 482 a - 4181\) , \( -12716 a + 119085\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(482a-4181\right){x}-12716a+119085$ |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$2.87064$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.601658500$ |
$41.02007025$ |
2.661320816 |
\( 6400 a - \frac{173888}{3} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 15 a + 202\) , \( 65 a + 688\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(15a+202\right){x}+65a+688$ |
| 9.1-b1 |
9.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$2.87064$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.601658500$ |
$41.02007025$ |
2.661320816 |
\( -6400 a - \frac{173888}{3} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -16 a + 202\) , \( -65 a + 688\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-16a+202\right){x}-65a+688$ |
| 9.1-c1 |
9.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$2.87064$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$8.112442038$ |
$9.761804657$ |
8.539501006 |
\( -6400 a - \frac{173888}{3} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -1248585 a - 11578838\) , \( -2342911510 a - 21727267427\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1248585a-11578838\right){x}-2342911510a-21727267427$ |
| 9.1-d1 |
9.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$2.87064$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$8.112442038$ |
$9.761804657$ |
8.539501006 |
\( 6400 a - \frac{173888}{3} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 1248584 a - 11578838\) , \( 2342911510 a - 21727267427\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1248584a-11578838\right){x}+2342911510a-21727267427$ |
| 10.1-a1 |
10.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{9} \cdot 5 \) |
$2.94726$ |
$(-11a-102), (a+9)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3^{2} \) |
$1$ |
$9.236270971$ |
4.481877208 |
\( -\frac{29803}{160} a - \frac{4197}{40} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( -11184 a - 103462\) , \( -2797470 a - 25941909\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-11184a-103462\right){x}-2797470a-25941909$ |
| 10.1-b1 |
10.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{9} \cdot 5 \) |
$2.94726$ |
$(-11a-102), (a+9)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$21.53246820$ |
1.160952880 |
\( -\frac{29803}{160} a - \frac{4197}{40} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -97 a + 921\) , \( 58 a - 557\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-97a+921\right){x}+58a-557$ |
| 10.2-a1 |
10.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{9} \cdot 5 \) |
$2.94726$ |
$(-11a-102), (a-9)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3^{2} \) |
$1$ |
$9.236270971$ |
4.481877208 |
\( \frac{29803}{160} a - \frac{4197}{40} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 11183 a - 103462\) , \( 2797470 a - 25941909\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(11183a-103462\right){x}+2797470a-25941909$ |
| 10.2-b1 |
10.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{86}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{9} \cdot 5 \) |
$2.94726$ |
$(-11a-102), (a-9)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$21.53246820$ |
1.160952880 |
\( \frac{29803}{160} a - \frac{4197}{40} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 97 a + 921\) , \( -58 a - 557\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(97a+921\right){x}-58a-557$ |
*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.
