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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 272734.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
272734.p1 | 272734p2 | \([1, 1, 0, -816, -9302]\) | \(10154161633/24334\) | \(144276286\) | \([]\) | \(142560\) | \(0.44483\) | |
272734.p2 | 272734p1 | \([1, 1, 0, -46, 92]\) | \(1877953/184\) | \(1090936\) | \([]\) | \(47520\) | \(-0.10448\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 272734.p have rank \(1\).
Complex multiplication
The elliptic curves in class 272734.p do not have complex multiplication.Modular form 272734.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.