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SageMath
E = EllipticCurve("hf1")
E.isogeny_class()
Elliptic curves in class 227850hf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
227850.c1 | 227850hf1 | \([1, 1, 0, -1822825, -1080402875]\) | \(-14575072995625/2522463552\) | \(-115923950948925000000\) | \([]\) | \(11197440\) | \(2.5761\) | \(\Gamma_0(N)\)-optimal |
227850.c2 | 227850hf2 | \([1, 1, 0, 12417800, 4635784000]\) | \(4607967053654375/2868211089408\) | \(-131813346272563200000000\) | \([]\) | \(33592320\) | \(3.1254\) |
Rank
sage: E.rank()
The elliptic curves in class 227850hf have rank \(1\).
Complex multiplication
The elliptic curves in class 227850hf do not have complex multiplication.Modular form 227850.2.a.hf
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.