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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 40425z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40425.e2 | 40425z1 | \([0, -1, 1, -3309868, -990379482]\) | \(1363413585016606720/644626239703677\) | \(1895990811872447384325\) | \([]\) | \(2534400\) | \(2.7770\) | \(\Gamma_0(N)\)-optimal |
| 40425.e1 | 40425z2 | \([0, -1, 1, -1065453958, 13386334297818]\) | \(116423188793017446400/91315917\) | \(104914319522783203125\) | \([]\) | \(12672000\) | \(3.5817\) |
Rank
sage: E.rank()
The elliptic curves in class 40425z have rank \(1\).
Complex multiplication
The elliptic curves in class 40425z do not have complex multiplication.Modular form 40425.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
