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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 72450.ek
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.ek1 | 72450el1 | \([1, -1, 1, -1162130, 482148497]\) | \(15238420194810961/12619514880\) | \(143744161680000000\) | \([2]\) | \(1290240\) | \(2.2206\) | \(\Gamma_0(N)\)-optimal |
72450.ek2 | 72450el2 | \([1, -1, 1, -910130, 696852497]\) | \(-7319577278195281/14169067365600\) | \(-161394532961287500000\) | \([2]\) | \(2580480\) | \(2.5672\) |
Rank
sage: E.rank()
The elliptic curves in class 72450.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.ek do not have complex multiplication.Modular form 72450.2.a.ek
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.