| Label |
Class |
Conductor |
Discriminant |
Rank* |
2-Selmer rank |
Torsion |
$\textrm{End}^0(J_{\overline\Q})$ |
$\textrm{End}^0(J)$ |
$\GL_2\textsf{-type}$ |
Sato-Tate |
Nonmaximal primes |
$\Q$-simple |
\(\overline{\Q}\)-simple |
\(\Aut(X)\) |
\(\Aut(X_{\overline{\Q}})\) |
$\Q$-points |
$\Q$-Weierstrass points |
mod-$\ell$ images |
Locally solvable |
Square Ш* |
Analytic Ш* |
Tamagawa |
Regulator |
Real period |
Leading coefficient |
Igusa-Clebsch invariants |
Igusa invariants |
G2-invariants |
Equation |
| 17689.a.17689.1 |
17689.a |
\( 7^{2} \cdot 19^{2} \) |
\( 7^{2} \cdot 19^{2} \) |
$0$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
\(\mathrm{M}_2(\Q)\) |
\(\mathsf{CM}\) |
✓ |
$E_6$ |
|
✓ |
|
$C_6$ |
$D_6$ |
$3$ |
$3$ |
2.240.1, 3.480.12 |
✓ |
✓ |
$1$ |
\( 1 \) |
\(1.000000\) |
\(15.833414\) |
\(0.989588\) |
$[580,11305,1902565,2264192]$ |
$[145,405,-395,-55325,17689]$ |
$[64097340625/17689,1234693125/17689,-8304875/17689]$ |
$y^2 + (x^2 + x)y = x^5 - 4x^4 + 2x^3 + x^2 - x$ |
| 17689.b.17689.1 |
17689.b |
\( 7^{2} \cdot 19^{2} \) |
\( 7^{2} \cdot 19^{2} \) |
$2$ |
$2$ |
$\mathsf{trivial}$ |
\(\mathrm{M}_2(\Q)\) |
\(\mathsf{CM}\) |
✓ |
$E_6$ |
|
✓ |
|
$C_6$ |
$D_6$ |
$12$ |
$0$ |
2.80.1, 3.480.12 |
✓ |
✓ |
$1$ |
\( 1 \) |
\(0.020782\) |
\(23.059848\) |
\(0.479221\) |
$[580,11305,1902565,2264192]$ |
$[145,405,-395,-55325,17689]$ |
$[64097340625/17689,1234693125/17689,-8304875/17689]$ |
$y^2 + (x^3 + x + 1)y = -3x^4 - 3x^3 + x^2 + x$ |
| 17689.c.17689.1 |
17689.c |
\( 7^{2} \cdot 19^{2} \) |
\( 7^{2} \cdot 19^{2} \) |
$2$ |
$2$ |
$\mathsf{trivial}$ |
\(\mathsf{RM}\) |
\(\mathsf{RM}\) |
✓ |
$\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
|
✓ |
✓ |
$C_2$ |
$C_2$ |
$8$ |
$0$ |
2.12.2, 3.2160.18 |
✓ |
✓ |
$1$ |
\( 1 \) |
\(0.028648\) |
\(15.318093\) |
\(0.438836\) |
$[92,7225,158147,-2264192]$ |
$[23,-279,-245,-20869,-17689]$ |
$[-6436343/17689,3394593/17689,2645/361]$ |
$y^2 + (x^3 + x + 1)y = 2x^5 + 2x^4 + x^3 + x^2$ |
| 17689.d.866761.1 |
17689.d |
\( 7^{2} \cdot 19^{2} \) |
\( 7^{4} \cdot 19^{2} \) |
$2$ |
$2$ |
$\Z/9\Z$ |
\(\mathsf{RM}\) |
\(\mathsf{RM}\) |
✓ |
$\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
|
✓ |
✓ |
$C_2$ |
$C_2$ |
$10$ |
$0$ |
2.12.2, 3.720.4 |
✓ |
✓ |
$1$ |
\( 3 \) |
\(1.596240\) |
\(11.235399\) |
\(0.664237\) |
$[3196,1064809,806830683,-110945408]$ |
$[799,-17767,-178217,-114515418,-866761]$ |
$[-325637113603999/866761,9062633983033/866761,113773911017/866761]$ |
$y^2 + (x^3 + x^2 + 1)y = 2x^4 - 2x^2 - 7x + 5$ |
| 17689.e.866761.1 |
17689.e |
\( 7^{2} \cdot 19^{2} \) |
\( 7^{4} \cdot 19^{2} \) |
$0$ |
$2$ |
$\Z/5\Z$ |
\(\mathsf{RM}\) |
\(\mathsf{RM}\) |
✓ |
$\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
|
✓ |
✓ |
$C_2$ |
$C_2$ |
$0$ |
$0$ |
2.12.2, 3.72.2 |
✓ |
✓ |
$4$ |
\( 5 \) |
\(1.000000\) |
\(2.783152\) |
\(2.226521\) |
$[82660,1,-22227071,110945408]$ |
$[20665,17793426,20428150568,26385430667561,866761]$ |
$[3768574004844424665625/866761,22431988071545220750/123823,178034344310076200/17689]$ |
$y^2 + (x^3 + x + 1)y = -x^6 - x^5 + 7x^4 + 7x^3 - 28x^2 - 13x + 34$ |
