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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 414050ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414050.ef2 | 414050ef1 | \([1, 0, 0, -211338, 1404969292]\) | \(-49/40\) | \(-852158796344025625000\) | \([]\) | \(20684160\) | \(2.6952\) | \(\Gamma_0(N)\)-optimal* |
414050.ef1 | 414050ef2 | \([1, 0, 0, -101653588, 394493688042]\) | \(-5452947409/250\) | \(-5325992477150160156250\) | \([]\) | \(62052480\) | \(3.2445\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414050ef have rank \(1\).
Complex multiplication
The elliptic curves in class 414050ef do not have complex multiplication.Modular form 414050.2.a.ef
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.