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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 25857e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25857.l2 | 25857e1 | \([1, -1, 0, -399294, 97127239]\) | \(2000852317801/2094417\) | \(7369716751682337\) | \([2]\) | \(387072\) | \(1.9620\) | \(\Gamma_0(N)\)-optimal |
25857.l1 | 25857e2 | \([1, -1, 0, -498159, 45420844]\) | \(3885442650361/1996623837\) | \(7025607669507630957\) | \([2]\) | \(774144\) | \(2.3085\) |
Rank
sage: E.rank()
The elliptic curves in class 25857e have rank \(0\).
Complex multiplication
The elliptic curves in class 25857e do not have complex multiplication.Modular form 25857.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.