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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 417450.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
417450.e1 | 417450e3 | \([1, 1, 0, -2542432400, 49287408672000]\) | \(65659235038126833886489/83360930070528000\) | \(2307483947448041472000000000\) | \([2]\) | \(418037760\) | \(4.1587\) | \(\Gamma_0(N)\)-optimal* |
417450.e2 | 417450e4 | \([1, 1, 0, -1860960400, 76307092000000]\) | \(-25748917201204045964569/75974386809024000000\) | \(-2103019697965333851000000000000\) | \([2]\) | \(836075520\) | \(4.5053\) | |
417450.e3 | 417450e1 | \([1, 1, 0, -121177025, -444830836875]\) | \(7108998764134921129/1026562059198720\) | \(28415895439939743780000000\) | \([2]\) | \(139345920\) | \(3.6094\) | \(\Gamma_0(N)\)-optimal* |
417450.e4 | 417450e2 | \([1, 1, 0, 200924975, -2406109914875]\) | \(32407784379748930391/109112951553764400\) | \(-3020316399492787722318750000\) | \([2]\) | \(278691840\) | \(3.9560\) |
Rank
sage: E.rank()
The elliptic curves in class 417450.e have rank \(1\).
Complex multiplication
The elliptic curves in class 417450.e do not have complex multiplication.Modular form 417450.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.