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 |
8.2-a1 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{4} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$91.38443470$ |
0.867070267 |
\( 8986661761218306693 a^{3} + 12242887281724452580 a^{2} - 7024786750071806011 a - 7608271795805318134 \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 2 a - 2\) , \( a^{3} - 4 a\) , \( -2 a^{3} - 9 a^{2} - 10 a + 2\) , \( 32 a^{3} + 44 a^{2} - 34 a - 39\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-2\right){x}^{2}+\left(-2a^{3}-9a^{2}-10a+2\right){x}+32a^{3}+44a^{2}-34a-39$ |
8.2-a2 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{11} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$182.7688694$ |
0.867070267 |
\( -601 a^{3} + 1330 a^{2} + 1931 a - 2180 \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 4 a\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} - a^{2} + 4 a + 5\) , \( -a^{3} - a^{2} + 4 a + 1\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-a^{3}-a^{2}+4a+5\right){x}-a^{3}-a^{2}+4a+1$ |
8.2-a3 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{8} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$365.5377388$ |
0.867070267 |
\( -2732527394 a^{3} + 1558959970 a^{2} + 13692770763 a + 7728253510 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 3 a - 2\) , \( 0\) , \( 9 a^{3} - 25 a^{2} + 14 a + 4\) , \( -30 a^{3} + 72 a^{2} + 25 a - 54\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{3}-3a-2\right){x}^{2}+\left(9a^{3}-25a^{2}+14a+4\right){x}-30a^{3}+72a^{2}+25a-54$ |
8.2-a4 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{10} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$91.38443470$ |
0.867070267 |
\( -63795770348435003481 a^{3} + 11114208719071878840 a^{2} + 264361023260378305267 a + 154509478780590380994 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 3 a - 2\) , \( 0\) , \( 154 a^{3} - 405 a^{2} + 39 a + 154\) , \( -2510 a^{3} + 6386 a^{2} + 264 a - 3160\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{3}-3a-2\right){x}^{2}+\left(154a^{3}-405a^{2}+39a+154\right){x}-2510a^{3}+6386a^{2}+264a-3160$ |
8.2-a5 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{10} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$365.5377388$ |
0.867070267 |
\( 2973144 a^{3} - 5002655 a^{2} - 8485568 a + 8881408 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( 3 a^{3} - 2 a^{2} - 9 a - 6\) , \( 6 a^{3} - 3 a^{2} - 22 a - 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a^{3}-2a^{2}-9a-6\right){x}+6a^{3}-3a^{2}-22a-13$ |
8.2-a6 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{11} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$45.69221735$ |
0.867070267 |
\( 34259927531436 a^{3} - 57543520518160 a^{2} - 97932989693671 a + 100819215025068 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( 18 a^{3} - 37 a^{2} - 14 a + 4\) , \( 63 a^{3} - 160 a^{2} - 5 a + 73\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(18a^{3}-37a^{2}-14a+4\right){x}+63a^{3}-160a^{2}-5a+73$ |
*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.