Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 37570j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37570.j4 | 37570j1 | \([1, -1, 1, -1933, -11339]\) | \(33076161/16640\) | \(401649148160\) | \([2]\) | \(40960\) | \(0.91925\) | \(\Gamma_0(N)\)-optimal |
| 37570.j2 | 37570j2 | \([1, -1, 1, -25053, -1518763]\) | \(72043225281/67600\) | \(1631699664400\) | \([2, 2]\) | \(81920\) | \(1.2658\) | |
| 37570.j3 | 37570j3 | \([1, -1, 1, -19273, -2242419]\) | \(-32798729601/71402500\) | \(-1723482770522500\) | \([2]\) | \(163840\) | \(1.6124\) | |
| 37570.j1 | 37570j4 | \([1, -1, 1, -400753, -97547683]\) | \(294889639316481/260\) | \(6275767940\) | \([2]\) | \(163840\) | \(1.6124\) |
Rank
sage: E.rank()
The elliptic curves in class 37570j have rank \(1\).
Complex multiplication
The elliptic curves in class 37570j do not have complex multiplication.Modular form 37570.2.a.j
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
