| 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 |
| 49.1-a1 |
49.1-a |
$2$ |
$3$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{12} \) |
$35.03996$ |
$(-a^4+6a^2+2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$273.4817282$ |
2.05850954 |
\( -\frac{3986622664}{117649} a^{4} + \frac{2668349896}{117649} a^{3} + \frac{20834045607}{117649} a^{2} - \frac{162996513}{16807} a - \frac{12309440397}{117649} \) |
\( \bigl[3 a^{4} - 2 a^{3} - 15 a^{2} + 2 a + 5\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{4} - a^{3} - 4 a^{2} + a + 1\) , \( -17 a^{4} + 92 a^{2} + 46 a - 22\) , \( 6 a^{4} - 2 a^{3} - 30 a^{2} - 11 a + 8\bigr] \) |
${y}^2+\left(3a^{4}-2a^{3}-15a^{2}+2a+5\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-17a^{4}+92a^{2}+46a-22\right){x}+6a^{4}-2a^{3}-30a^{2}-11a+8$ |
| 49.1-a2 |
49.1-a |
$2$ |
$3$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{8} \) |
$35.03996$ |
$(-a^4+6a^2+2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$273.4817282$ |
2.05850954 |
\( \frac{392767}{49} a^{4} + \frac{404875}{49} a^{3} - \frac{2995121}{49} a^{2} - \frac{227585}{7} a + \frac{856140}{49} \) |
\( \bigl[3 a^{4} - 2 a^{3} - 15 a^{2} + a + 5\) , \( -3 a^{4} + 2 a^{3} + 15 a^{2} - 6\) , \( 3 a^{4} - 2 a^{3} - 15 a^{2} + a + 5\) , \( -2 a^{4} + a^{3} + 8 a^{2} + 3 a + 3\) , \( -11 a^{4} + 7 a^{3} + 56 a^{2} + a - 26\bigr] \) |
${y}^2+\left(3a^{4}-2a^{3}-15a^{2}+a+5\right){x}{y}+\left(3a^{4}-2a^{3}-15a^{2}+a+5\right){y}={x}^{3}+\left(-3a^{4}+2a^{3}+15a^{2}-6\right){x}^{2}+\left(-2a^{4}+a^{3}+8a^{2}+3a+3\right){x}-11a^{4}+7a^{3}+56a^{2}+a-26$ |
| 49.1-b1 |
49.1-b |
$2$ |
$3$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{2} \) |
$35.03996$ |
$(-a^4+6a^2+2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$336.2215358$ |
1.26537748 |
\( -32590851870 a^{4} + 22201278229 a^{3} + 170088446312 a^{2} - 10951938243 a - 101553473094 \) |
\( \bigl[a^{4} - 6 a^{2} - 2 a + 3\) , \( -a^{4} + a^{3} + 4 a^{2} + 1\) , \( 2 a^{4} - a^{3} - 11 a^{2} + 6\) , \( -11 a^{4} + 16 a^{3} + 42 a^{2} - 35 a + 3\) , \( -89 a^{4} + 160 a^{3} + 310 a^{2} - 419 a + 92\bigr] \) |
${y}^2+\left(a^{4}-6a^{2}-2a+3\right){x}{y}+\left(2a^{4}-a^{3}-11a^{2}+6\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}+1\right){x}^{2}+\left(-11a^{4}+16a^{3}+42a^{2}-35a+3\right){x}-89a^{4}+160a^{3}+310a^{2}-419a+92$ |
| 49.1-b2 |
49.1-b |
$2$ |
$3$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{2} \) |
$35.03996$ |
$(-a^4+6a^2+2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$336.2215358$ |
1.26537748 |
\( -78417003901 a^{4} + 219037828056 a^{3} - 703259394 a^{2} - 155572879252 a + 43729126161 \) |
\( \bigl[3 a^{4} - 2 a^{3} - 15 a^{2} + a + 6\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a - 1\) , \( 2 a^{4} - a^{3} - 11 a^{2} + a + 6\) , \( a^{4} + 2 a^{3} - 8 a^{2} - 9 a - 1\) , \( -12 a^{4} + 4 a^{3} + 59 a^{2} + 22 a - 16\bigr] \) |
${y}^2+\left(3a^{4}-2a^{3}-15a^{2}+a+6\right){x}{y}+\left(2a^{4}-a^{3}-11a^{2}+a+6\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+2a-1\right){x}^{2}+\left(a^{4}+2a^{3}-8a^{2}-9a-1\right){x}-12a^{4}+4a^{3}+59a^{2}+22a-16$ |
| 49.1-c1 |
49.1-c |
$2$ |
$3$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$35.03996$ |
$(-a^4+6a^2+2a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1.276891735$ |
$116.8049831$ |
2.80659680 |
\( -32590851870 a^{4} + 22201278229 a^{3} + 170088446312 a^{2} - 10951938243 a - 101553473094 \) |
\( \bigl[a\) , \( -a^{4} + 5 a^{2} + 2 a\) , \( 3 a^{4} - 2 a^{3} - 15 a^{2} + 2 a + 5\) , \( -6 a^{4} + 16 a^{3} + 18 a^{2} - 53 a\) , \( -63 a^{4} + 161 a^{3} + 176 a^{2} - 503 a + 108\bigr] \) |
${y}^2+a{x}{y}+\left(3a^{4}-2a^{3}-15a^{2}+2a+5\right){y}={x}^{3}+\left(-a^{4}+5a^{2}+2a\right){x}^{2}+\left(-6a^{4}+16a^{3}+18a^{2}-53a\right){x}-63a^{4}+161a^{3}+176a^{2}-503a+108$ |
| 49.1-c2 |
49.1-c |
$2$ |
$3$ |
5.5.70601.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$35.03996$ |
$(-a^4+6a^2+2a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$0.425630578$ |
$350.4149495$ |
2.80659680 |
\( -78417003901 a^{4} + 219037828056 a^{3} - 703259394 a^{2} - 155572879252 a + 43729126161 \) |
\( \bigl[2 a^{4} - 2 a^{3} - 9 a^{2} + 3 a + 3\) , \( -2 a^{4} + a^{3} + 10 a^{2} + a - 4\) , \( 0\) , \( -15 a^{4} + 12 a^{3} + 74 a^{2} - 8 a - 40\) , \( -33 a^{4} + 21 a^{3} + 176 a^{2} - 10 a - 106\bigr] \) |
${y}^2+\left(2a^{4}-2a^{3}-9a^{2}+3a+3\right){x}{y}={x}^{3}+\left(-2a^{4}+a^{3}+10a^{2}+a-4\right){x}^{2}+\left(-15a^{4}+12a^{3}+74a^{2}-8a-40\right){x}-33a^{4}+21a^{3}+176a^{2}-10a-106$ |
*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.
