Label |
Cremona label |
Class |
Cremona class |
Class size |
Class degree |
Conductor |
Discriminant |
Rank |
Torsion |
$\textrm{End}^0(E_{\overline\Q})$ |
CM |
Sato-Tate |
Semistable |
Potentially good |
Nonmax $\ell$ |
$\ell$-adic images |
mod-$\ell$ images |
Adelic level |
Adelic index |
Adelic genus |
Regulator |
$Ш_{\textrm{an}}$ |
Ш primes |
Integral points |
Modular degree |
Faltings height |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
mod-$m$ images |
MW-generators |
57.a1 |
57a1 |
57.a |
57a |
$1$ |
$1$ |
\( 3 \cdot 19 \) |
\( - 3^{2} \cdot 19 \) |
$1$ |
$\mathsf{trivial}$ |
$\Q$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
$38$ |
$2$ |
$0$ |
$0.037574592$ |
$1$ |
|
$12$ |
$4$ |
$-0.836549$ |
$-1404928/171$ |
$[0, -1, 1, -2, 2]$ |
\(y^2+y=x^3-x^2-2x+2\) |
38.2.0.a.1 |
$[(2, 1)]$ |
57.b1 |
57c2 |
57.b |
57c |
$2$ |
$5$ |
\( 3 \cdot 19 \) |
\( - 3^{2} \cdot 19^{5} \) |
$0$ |
$\mathsf{trivial}$ |
$\Q$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
$5$ |
5.24.0.3 |
5B.1.2 |
$190$ |
$48$ |
$1$ |
$1$ |
$1$ |
|
$0$ |
$60$ |
$0.651407$ |
$-9358714467168256/22284891$ |
$[0, 1, 1, -4390, -113432]$ |
\(y^2+y=x^3+x^2-4390x-113432\) |
5.24.0-5.a.2.2, 38.2.0.a.1, 190.48.1.? |
$[]$ |
57.b2 |
57c1 |
57.b |
57c |
$2$ |
$5$ |
\( 3 \cdot 19 \) |
\( - 3^{10} \cdot 19 \) |
$0$ |
$\Z/5\Z$ |
$\Q$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
$5$ |
5.24.0.1 |
5B.1.1 |
$190$ |
$48$ |
$1$ |
$1$ |
$1$ |
|
$4$ |
$12$ |
$-0.153312$ |
$841232384/1121931$ |
$[0, 1, 1, 20, -32]$ |
\(y^2+y=x^3+x^2+20x-32\) |
5.24.0-5.a.1.2, 38.2.0.a.1, 190.48.1.? |
$[]$ |
57.c1 |
57b3 |
57.c |
57b |
$4$ |
$4$ |
\( 3 \cdot 19 \) |
\( 3^{4} \cdot 19 \) |
$0$ |
$\Z/4\Z$ |
$\Q$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
$2$ |
4.12.0.7 |
2B |
$456$ |
$48$ |
$0$ |
$1$ |
$1$ |
|
$2$ |
$6$ |
$-0.245009$ |
$115714886617/1539$ |
$[1, 0, 1, -102, 385]$ |
\(y^2+xy+y=x^3-102x+385\) |
2.3.0.a.1, 4.12.0-4.c.1.1, 24.24.0-24.z.1.8, 76.24.0.?, 456.48.0.? |
$[]$ |
57.c2 |
57b1 |
57.c |
57b |
$4$ |
$4$ |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\Q$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
$2$ |
4.12.0.1 |
2Cs |
$228$ |
$48$ |
$0$ |
$1$ |
$1$ |
|
$2$ |
$3$ |
$-0.591582$ |
$30664297/3249$ |
$[1, 0, 1, -7, 5]$ |
\(y^2+xy+y=x^3-7x+5\) |
2.6.0.a.1, 4.12.0-2.a.1.1, 12.24.0-12.b.1.2, 76.24.0.?, 228.48.0.? |
$[]$ |
57.c3 |
57b2 |
57.c |
57b |
$4$ |
$4$ |
\( 3 \cdot 19 \) |
\( 3 \cdot 19 \) |
$0$ |
$\Z/2\Z$ |
$\Q$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
$2$ |
8.12.0.6 |
2B |
$456$ |
$48$ |
$0$ |
$1$ |
$1$ |
|
$1$ |
$6$ |
$-0.938156$ |
$389017/57$ |
$[1, 0, 1, -2, -1]$ |
\(y^2+xy+y=x^3-2x-1\) |
2.3.0.a.1, 4.6.0.c.1, 8.12.0-4.c.1.5, 12.12.0-4.c.1.2, 24.24.0-24.z.1.4, $\ldots$ |
$[]$ |
57.c4 |
57b4 |
57.c |
57b |
$4$ |
$4$ |
\( 3 \cdot 19 \) |
\( - 3 \cdot 19^{4} \) |
$0$ |
$\Z/2\Z$ |
$\Q$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
$2$ |
4.12.0.8 |
2B |
$456$ |
$48$ |
$0$ |
$1$ |
$1$ |
|
$0$ |
$6$ |
$-0.245009$ |
$67419143/390963$ |
$[1, 0, 1, 8, 29]$ |
\(y^2+xy+y=x^3+8x+29\) |
2.3.0.a.1, 4.12.0-4.c.1.2, 6.6.0.a.1, 12.24.0-12.g.1.1, 152.24.0.?, $\ldots$ |
$[]$ |