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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 49725p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
49725.v2 | 49725p1 | \([1, -1, 0, -59067, 5535216]\) | \(2000852317801/2094417\) | \(23856718640625\) | \([2]\) | \(184320\) | \(1.4842\) | \(\Gamma_0(N)\)-optimal |
49725.v1 | 49725p2 | \([1, -1, 0, -73692, 2595591]\) | \(3885442650361/1996623837\) | \(22742793393328125\) | \([2]\) | \(368640\) | \(1.8308\) |
Rank
sage: E.rank()
The elliptic curves in class 49725p have rank \(0\).
Complex multiplication
The elliptic curves in class 49725p do not have complex multiplication.Modular form 49725.2.a.p
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.