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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 40080.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40080.p1 | 40080v1 | \([0, -1, 0, -125, 0]\) | \(13608288256/7828125\) | \(125250000\) | \([2]\) | \(11232\) | \(0.24342\) | \(\Gamma_0(N)\)-optimal |
| 40080.p2 | 40080v2 | \([0, -1, 0, 500, -500]\) | \(53892071984/31375125\) | \(-8032032000\) | \([2]\) | \(22464\) | \(0.58999\) |
Rank
sage: E.rank()
The elliptic curves in class 40080.p have rank \(0\).
Complex multiplication
The elliptic curves in class 40080.p do not have complex multiplication.Modular form 40080.2.a.p
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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
