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SageMath
E = EllipticCurve("ev1")
E.isogeny_class()
Elliptic curves in class 333270.ev
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.ev1 | 333270ev2 | \([1, -1, 1, -178292, 25468759]\) | \(70665260180607/9411920000\) | \(83481311536560000\) | \([2]\) | \(4128768\) | \(1.9747\) | |
333270.ev2 | 333270ev1 | \([1, -1, 1, -45812, -3358889]\) | \(1198785674367/140492800\) | \(1246135029350400\) | \([2]\) | \(2064384\) | \(1.6281\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 333270.ev have rank \(2\).
Complex multiplication
The elliptic curves in class 333270.ev do not have complex multiplication.Modular form 333270.2.a.ev
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 LMFDB 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 LMFDB labels.