| 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 |
| 43.2-a1 |
43.2-a |
$2$ |
$7$ |
5.5.38569.1 |
$5$ |
$[5, 0]$ |
43.2 |
\( 43 \) |
\( 43^{7} \) |
$25.56257$ |
$(-a^4+6a^2+2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$49$ |
\( 7 \) |
$1$ |
$0.856262961$ |
1.49548510 |
\( \frac{503430677683141400412260203661}{271818611107} a^{4} - \frac{983743624658765679781695980679}{271818611107} a^{3} - \frac{594840025499343907143854406374}{271818611107} a^{2} + \frac{1162364767809040613354336495114}{271818611107} a - \frac{257630586761460389878355097065}{271818611107} \) |
\( \bigl[a^{4} - 4 a^{2} + 1\) , \( a^{4} - a^{3} - 4 a^{2} + 5 a\) , \( -a^{4} + 5 a^{2} - 3\) , \( -438 a^{4} - 108 a^{3} + 2123 a^{2} + 558 a - 1557\) , \( -6146 a^{4} - 1666 a^{3} + 30140 a^{2} + 8257 a - 22182\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+1\right){x}{y}+\left(-a^{4}+5a^{2}-3\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+5a\right){x}^{2}+\left(-438a^{4}-108a^{3}+2123a^{2}+558a-1557\right){x}-6146a^{4}-1666a^{3}+30140a^{2}+8257a-22182$ |
| 43.2-a2 |
43.2-a |
$2$ |
$7$ |
5.5.38569.1 |
$5$ |
$[5, 0]$ |
43.2 |
\( 43 \) |
\( 43 \) |
$25.56257$ |
$(-a^4+6a^2+2a-3)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$14391.21159$ |
1.49548510 |
\( \frac{76659}{43} a^{4} + \frac{48039}{43} a^{3} - \frac{312145}{43} a^{2} - \frac{219027}{43} a + \frac{79541}{43} \) |
\( \bigl[a^{3} - 3 a + 1\) , \( a^{4} - 4 a^{2} - a + 2\) , \( a^{2} + a - 1\) , \( -a + 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-3a+1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{4}-4a^{2}-a+2\right){x}^{2}+\left(-a+1\right){x}$ |
| 43.2-b1 |
43.2-b |
$1$ |
$1$ |
5.5.38569.1 |
$5$ |
$[5, 0]$ |
43.2 |
\( 43 \) |
\( 43 \) |
$25.56257$ |
$(-a^4+6a^2+2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$338.5842133$ |
1.72404071 |
\( \frac{434298794}{43} a^{4} + \frac{343516845}{43} a^{3} - \frac{1899978901}{43} a^{2} - \frac{1502757203}{43} a + \frac{549334398}{43} \) |
\( \bigl[-a^{4} + 5 a^{2} - 3\) , \( a^{2} + a - 1\) , \( a^{4} - 4 a^{2} + a + 2\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 4\) , \( -3 a^{4} + 5 a^{3} + 4 a^{2} - 5 a\bigr] \) |
${y}^2+\left(-a^{4}+5a^{2}-3\right){x}{y}+\left(a^{4}-4a^{2}+a+2\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(a^{4}+a^{3}-4a^{2}-3a+4\right){x}-3a^{4}+5a^{3}+4a^{2}-5a$ |
| 43.2-c1 |
43.2-c |
$1$ |
$1$ |
5.5.38569.1 |
$5$ |
$[5, 0]$ |
43.2 |
\( 43 \) |
\( 43 \) |
$25.56257$ |
$(-a^4+6a^2+2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$286.1746394$ |
1.45717582 |
\( -\frac{51665932627}{43} a^{4} - \frac{10018445802}{43} a^{3} + \frac{250693585412}{43} a^{2} + \frac{53592431783}{43} a - \frac{173974818865}{43} \) |
\( \bigl[a^{3} - 4 a\) , \( -3 a^{4} - a^{3} + 14 a^{2} + 4 a - 7\) , \( -a^{4} + 5 a^{2} - 2\) , \( -2 a^{4} - 3 a^{3} + 4 a^{2} + 5 a + 3\) , \( 2 a^{4} + 5 a^{3} + a^{2} - 3 a - 2\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(-a^{4}+5a^{2}-2\right){y}={x}^{3}+\left(-3a^{4}-a^{3}+14a^{2}+4a-7\right){x}^{2}+\left(-2a^{4}-3a^{3}+4a^{2}+5a+3\right){x}+2a^{4}+5a^{3}+a^{2}-3a-2$ |
| 43.2-d1 |
43.2-d |
$1$ |
$1$ |
5.5.38569.1 |
$5$ |
$[5, 0]$ |
43.2 |
\( 43 \) |
\( 43 \) |
$25.56257$ |
$(-a^4+6a^2+2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.006318915$ |
$12588.52134$ |
2.02520088 |
\( -\frac{51665932627}{43} a^{4} - \frac{10018445802}{43} a^{3} + \frac{250693585412}{43} a^{2} + \frac{53592431783}{43} a - \frac{173974818865}{43} \) |
\( \bigl[2 a^{4} + a^{3} - 9 a^{2} - 3 a + 5\) , \( 2 a^{4} - 9 a^{2} + 3\) , \( a^{2} + a - 2\) , \( 6 a^{4} + 2 a^{3} - 27 a^{2} - 14 a + 18\) , \( -9 a^{4} + 14 a^{3} + 12 a^{2} - 10 a + 6\bigr] \) |
${y}^2+\left(2a^{4}+a^{3}-9a^{2}-3a+5\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(2a^{4}-9a^{2}+3\right){x}^{2}+\left(6a^{4}+2a^{3}-27a^{2}-14a+18\right){x}-9a^{4}+14a^{3}+12a^{2}-10a+6$ |
| 43.2-e1 |
43.2-e |
$1$ |
$1$ |
5.5.38569.1 |
$5$ |
$[5, 0]$ |
43.2 |
\( 43 \) |
\( 43 \) |
$25.56257$ |
$(-a^4+6a^2+2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.008342728$ |
$9825.480120$ |
2.08695326 |
\( \frac{434298794}{43} a^{4} + \frac{343516845}{43} a^{3} - \frac{1899978901}{43} a^{2} - \frac{1502757203}{43} a + \frac{549334398}{43} \) |
\( \bigl[a^{2} + a - 2\) , \( -1\) , \( a^{4} + a^{3} - 5 a^{2} - 3 a + 4\) , \( a^{4} - 5 a^{2} - a + 3\) , \( -2 a^{3} + 3 a^{2} + 3 a - 3\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{4}+a^{3}-5a^{2}-3a+4\right){y}={x}^{3}-{x}^{2}+\left(a^{4}-5a^{2}-a+3\right){x}-2a^{3}+3a^{2}+3a-3$ |
| 43.2-f1 |
43.2-f |
$2$ |
$7$ |
5.5.38569.1 |
$5$ |
$[5, 0]$ |
43.2 |
\( 43 \) |
\( 43^{7} \) |
$25.56257$ |
$(-a^4+6a^2+2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.6.3 |
$1$ |
\( 1 \) |
$0.453019257$ |
$172.5067413$ |
1.98963563 |
\( \frac{503430677683141400412260203661}{271818611107} a^{4} - \frac{983743624658765679781695980679}{271818611107} a^{3} - \frac{594840025499343907143854406374}{271818611107} a^{2} + \frac{1162364767809040613354336495114}{271818611107} a - \frac{257630586761460389878355097065}{271818611107} \) |
\( \bigl[-a^{4} + 5 a^{2} + a - 2\) , \( a^{4} + a^{3} - 5 a^{2} - 4 a + 3\) , \( 2 a^{4} + a^{3} - 9 a^{2} - 3 a + 4\) , \( -15 a^{4} - 5 a^{3} + 79 a^{2} + 29 a - 122\) , \( 27 a^{4} - 24 a^{3} - 192 a^{2} + 18 a + 392\bigr] \) |
${y}^2+\left(-a^{4}+5a^{2}+a-2\right){x}{y}+\left(2a^{4}+a^{3}-9a^{2}-3a+4\right){y}={x}^{3}+\left(a^{4}+a^{3}-5a^{2}-4a+3\right){x}^{2}+\left(-15a^{4}-5a^{3}+79a^{2}+29a-122\right){x}+27a^{4}-24a^{3}-192a^{2}+18a+392$ |
| 43.2-f2 |
43.2-f |
$2$ |
$7$ |
5.5.38569.1 |
$5$ |
$[5, 0]$ |
43.2 |
\( 43 \) |
\( 43 \) |
$25.56257$ |
$(-a^4+6a^2+2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.6.1 |
$1$ |
\( 1 \) |
$0.064717036$ |
$1207.547189$ |
1.98963563 |
\( \frac{76659}{43} a^{4} + \frac{48039}{43} a^{3} - \frac{312145}{43} a^{2} - \frac{219027}{43} a + \frac{79541}{43} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 5 a + 1\) , \( 0\) , \( -a^{4} + 5 a^{2} + 3 a + 1\) , \( a^{2} + 2 a\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(-a^{4}+5a^{2}+3a+1\right){x}+a^{2}+2a$ |
*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.
