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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 325e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
325.a2 | 325e1 | \([0, -1, 1, -98, 378]\) | \(4206161920/371293\) | \(9282325\) | \([5]\) | \(84\) | \(0.077191\) | \(\Gamma_0(N)\)-optimal |
325.a1 | 325e2 | \([0, -1, 1, -12708, -547182]\) | \(23242854400/13\) | \(126953125\) | \([]\) | \(420\) | \(0.88191\) |
Rank
sage: E.rank()
The elliptic curves in class 325e have rank \(0\).
Complex multiplication
The elliptic curves in class 325e do not have complex multiplication.Modular form 325.2.a.e
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.