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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 259182eu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259182.eu2 | 259182eu1 | \([1, -1, 1, -19625, -1048539]\) | \(647214625/3332\) | \(4303171272708\) | \([2]\) | \(537600\) | \(1.2701\) | \(\Gamma_0(N)\)-optimal |
259182.eu1 | 259182eu2 | \([1, -1, 1, -30515, 253905]\) | \(2433138625/1387778\) | \(1792270835082882\) | \([2]\) | \(1075200\) | \(1.6166\) |
Rank
sage: E.rank()
The elliptic curves in class 259182eu have rank \(1\).
Complex multiplication
The elliptic curves in class 259182eu do not have complex multiplication.Modular form 259182.2.a.eu
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.