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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 36414z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
36414.a2 | 36414z1 | \([1, -1, 0, -156114, -2746796]\) | \(23912763841/13647872\) | \(240151883583209472\) | \([2]\) | \(774144\) | \(2.0248\) | \(\Gamma_0(N)\)-optimal |
36414.a1 | 36414z2 | \([1, -1, 0, -1820754, -943268396]\) | \(37936442980801/88817792\) | \(1562863429881355392\) | \([2]\) | \(1548288\) | \(2.3714\) |
Rank
sage: E.rank()
The elliptic curves in class 36414z have rank \(2\).
Complex multiplication
The elliptic curves in class 36414z do not have complex multiplication.Modular form 36414.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.