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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 525.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
525.c1 | 525c1 | \([1, 1, 0, -450, 3375]\) | \(5177717/189\) | \(369140625\) | \([2]\) | \(240\) | \(0.41350\) | \(\Gamma_0(N)\)-optimal |
525.c2 | 525c2 | \([1, 1, 0, 175, 12750]\) | \(300763/35721\) | \(-69767578125\) | \([2]\) | \(480\) | \(0.76007\) |
Rank
sage: E.rank()
The elliptic curves in class 525.c have rank \(1\).
Complex multiplication
The elliptic curves in class 525.c do not have complex multiplication.Modular form 525.2.a.c
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.