| 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 |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$25.19340$ |
$(a^4-2a^3-4a^2+6a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.065885382$ |
$2381.844613$ |
2.05359085 |
\( \frac{19567537797912176446}{2401} a^{4} - \frac{69587777857668838427}{2401} a^{3} + \frac{49595930261394340581}{2401} a^{2} + \frac{20652200646025945743}{2401} a - \frac{12573218933743386648}{2401} \) |
\( \bigl[2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 4\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 2\) , \( a^{2} - 2\) , \( -127 a^{4} + 191 a^{3} + 470 a^{2} - 400 a - 351\) , \( -1617 a^{4} + 2597 a^{3} + 5876 a^{2} - 5786 a - 3783\bigr] \) |
${y}^2+\left(2a^{4}-3a^{3}-7a^{2}+7a+4\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{4}-2a^{3}-3a^{2}+6a+2\right){x}^{2}+\left(-127a^{4}+191a^{3}+470a^{2}-400a-351\right){x}-1617a^{4}+2597a^{3}+5876a^{2}-5786a-3783$ |
| 49.1-a2 |
49.1-a |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{2} \) |
$25.19340$ |
$(a^4-2a^3-4a^2+6a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.016471345$ |
$19054.75690$ |
2.05359085 |
\( \frac{827}{7} a^{4} + 5033 a^{3} - \frac{28282}{7} a^{2} - \frac{26855}{7} a + \frac{13946}{7} \) |
\( \bigl[a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 2\) , \( 2 a^{4} - 3 a^{3} - 8 a^{2} + 9 a + 5\) , \( 2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 2\) , \( 10 a^{4} - 17 a^{3} - 35 a^{2} + 39 a + 24\) , \( 16 a^{4} - 27 a^{3} - 54 a^{2} + 56 a + 36\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-3a^{2}+6a+2\right){x}{y}+\left(2a^{4}-3a^{3}-6a^{2}+7a+2\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-8a^{2}+9a+5\right){x}^{2}+\left(10a^{4}-17a^{3}-35a^{2}+39a+24\right){x}+16a^{4}-27a^{3}-54a^{2}+56a+36$ |
| 49.1-a3 |
49.1-a |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{4} \) |
$25.19340$ |
$(a^4-2a^3-4a^2+6a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.032942691$ |
$19054.75690$ |
2.05359085 |
\( \frac{4128570706}{49} a^{4} - \frac{2238511927}{49} a^{3} - \frac{4911331220}{49} a^{2} + \frac{180417913}{49} a + \frac{791318512}{49} \) |
\( \bigl[2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 4\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 2\) , \( a^{2} - 2\) , \( -32 a^{4} + 46 a^{3} + 115 a^{2} - 100 a - 76\) , \( 40 a^{4} - 60 a^{3} - 144 a^{2} + 132 a + 97\bigr] \) |
${y}^2+\left(2a^{4}-3a^{3}-7a^{2}+7a+4\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{4}-2a^{3}-3a^{2}+6a+2\right){x}^{2}+\left(-32a^{4}+46a^{3}+115a^{2}-100a-76\right){x}+40a^{4}-60a^{3}-144a^{2}+132a+97$ |
| 49.1-a4 |
49.1-a |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{2} \) |
$25.19340$ |
$(a^4-2a^3-4a^2+6a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.065885382$ |
$4763.689226$ |
2.05359085 |
\( \frac{620948251468503566}{7} a^{4} + 28210247096660731 a^{3} - \frac{1405103809967670307}{7} a^{2} - \frac{152315338627341705}{7} a + \frac{267878898946671192}{7} \) |
\( \bigl[a + 1\) , \( a^{4} - a^{3} - 3 a^{2} + 1\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( 9 a^{4} + 37 a^{3} - 134 a^{2} - 23 a + 26\) , \( -112 a^{4} - 62 a^{3} + 868 a^{2} - 258 a - 38\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(2a^{4}-3a^{3}-7a^{2}+7a+3\right){y}={x}^{3}+\left(a^{4}-a^{3}-3a^{2}+1\right){x}^{2}+\left(9a^{4}+37a^{3}-134a^{2}-23a+26\right){x}-112a^{4}-62a^{3}+868a^{2}-258a-38$ |
| 49.1-b1 |
49.1-b |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$25.19340$ |
$(a^4-2a^3-4a^2+6a+2)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$14.54729216$ |
1.21835419 |
\( \frac{19567537797912176446}{2401} a^{4} - \frac{69587777857668838427}{2401} a^{3} + \frac{49595930261394340581}{2401} a^{2} + \frac{20652200646025945743}{2401} a - \frac{12573218933743386648}{2401} \) |
\( \bigl[2 a^{4} - 3 a^{3} - 6 a^{2} + 6 a + 1\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( 2 a^{4} - 3 a^{3} + 5 a^{2} + a - 44\) , \( 20 a^{4} - 15 a^{3} - 37 a^{2} - 26 a - 133\bigr] \) |
${y}^2+\left(2a^{4}-3a^{3}-6a^{2}+6a+1\right){x}{y}+\left(a^{4}-2a^{3}-3a^{2}+6a+1\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){x}^{2}+\left(2a^{4}-3a^{3}+5a^{2}+a-44\right){x}+20a^{4}-15a^{3}-37a^{2}-26a-133$ |
| 49.1-b2 |
49.1-b |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{2} \) |
$25.19340$ |
$(a^4-2a^3-4a^2+6a+2)$ |
$0 \le r \le 2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$3724.106794$ |
1.21835419 |
\( \frac{827}{7} a^{4} + 5033 a^{3} - \frac{28282}{7} a^{2} - \frac{26855}{7} a + \frac{13946}{7} \) |
\( \bigl[2 a^{4} - 3 a^{3} - 6 a^{2} + 6 a + 1\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( 2 a^{4} - 3 a^{3} - 5 a^{2} + 6 a + 1\) , \( 2 a^{4} - 2 a^{3} - 7 a^{2} + 4 a + 2\bigr] \) |
${y}^2+\left(2a^{4}-3a^{3}-6a^{2}+6a+1\right){x}{y}+\left(a^{4}-2a^{3}-3a^{2}+6a+1\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){x}^{2}+\left(2a^{4}-3a^{3}-5a^{2}+6a+1\right){x}+2a^{4}-2a^{3}-7a^{2}+4a+2$ |
| 49.1-b3 |
49.1-b |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{4} \) |
$25.19340$ |
$(a^4-2a^3-4a^2+6a+2)$ |
$0 \le r \le 2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$465.5133493$ |
1.21835419 |
\( \frac{4128570706}{49} a^{4} - \frac{2238511927}{49} a^{3} - \frac{4911331220}{49} a^{2} + \frac{180417913}{49} a + \frac{791318512}{49} \) |
\( \bigl[2 a^{4} - 3 a^{3} - 6 a^{2} + 6 a + 1\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( 7 a^{4} - 3 a^{3} - 25 a^{2} - 9 a - 4\) , \( 32 a^{4} - 23 a^{3} - 123 a^{2} + 22\bigr] \) |
${y}^2+\left(2a^{4}-3a^{3}-6a^{2}+6a+1\right){x}{y}+\left(a^{4}-2a^{3}-3a^{2}+6a+1\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){x}^{2}+\left(7a^{4}-3a^{3}-25a^{2}-9a-4\right){x}+32a^{4}-23a^{3}-123a^{2}+22$ |
| 49.1-b4 |
49.1-b |
$4$ |
$4$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{2} \) |
$25.19340$ |
$(a^4-2a^3-4a^2+6a+2)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$58.18916866$ |
1.21835419 |
\( \frac{620948251468503566}{7} a^{4} + 28210247096660731 a^{3} - \frac{1405103809967670307}{7} a^{2} - \frac{152315338627341705}{7} a + \frac{267878898946671192}{7} \) |
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 3 a + 3\) , \( -2 a^{4} + 3 a^{3} + 7 a^{2} - 8 a - 2\) , \( 0\) , \( -15 a^{4} + 15 a^{3} + 36 a^{2} - 32 a - 19\) , \( -15 a^{4} - 48 a^{3} - 18 a^{2} + 125 a + 13\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-4a^{2}+3a+3\right){x}{y}={x}^{3}+\left(-2a^{4}+3a^{3}+7a^{2}-8a-2\right){x}^{2}+\left(-15a^{4}+15a^{3}+36a^{2}-32a-19\right){x}-15a^{4}-48a^{3}-18a^{2}+125a+13$ |
*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.
