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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 21294n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21294.j2 | 21294n1 | \([1, -1, 0, -255813, -92353195]\) | \(-8869743/12544\) | \(-2618288562375472896\) | \([2]\) | \(419328\) | \(2.2263\) | \(\Gamma_0(N)\)-optimal |
21294.j1 | 21294n2 | \([1, -1, 0, -5001333, -4301629435]\) | \(66282611823/38416\) | \(8018508722274885744\) | \([2]\) | \(838656\) | \(2.5729\) |
Rank
sage: E.rank()
The elliptic curves in class 21294n have rank \(1\).
Complex multiplication
The elliptic curves in class 21294n do not have complex multiplication.Modular form 21294.2.a.n
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.