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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 414050z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414050.z2 | 414050z1 | \([1, 1, 0, 927300, -287981000]\) | \(397535/392\) | \(-86954979218778125000\) | \([]\) | \(13271040\) | \(2.5130\) | \(\Gamma_0(N)\)-optimal* |
414050.z1 | 414050z2 | \([1, 1, 0, -9423950, 15642592750]\) | \(-417267265/235298\) | \(-52194726276071569531250\) | \([]\) | \(39813120\) | \(3.0623\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414050z have rank \(1\).
Complex multiplication
The elliptic curves in class 414050z do not have complex multiplication.Modular form 414050.2.a.z
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.