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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 490245o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
490245.o4 | 490245o1 | \([1, 0, 0, -257251, 48856856]\) | \(16003198512756001/488525390625\) | \(57474523681640625\) | \([2]\) | \(4718592\) | \(1.9918\) | \(\Gamma_0(N)\)-optimal* |
490245.o2 | 490245o2 | \([1, 0, 0, -4085376, 3177966231]\) | \(64096096056024006001/62562515625\) | \(7360417400765625\) | \([2, 2]\) | \(9437184\) | \(2.3384\) | \(\Gamma_0(N)\)-optimal* |
490245.o1 | 490245o3 | \([1, 0, 0, -65366001, 203406280356]\) | \(262537424941059264096001/250125\) | \(29426956125\) | \([2]\) | \(18874368\) | \(2.6850\) | \(\Gamma_0(N)\)-optimal* |
490245.o3 | 490245o4 | \([1, 0, 0, -4054751, 3227964606]\) | \(-62665433378363916001/2004003001000125\) | \(-235768949064663706125\) | \([2]\) | \(18874368\) | \(2.6850\) |
Rank
sage: E.rank()
The elliptic curves in class 490245o have rank \(2\).
Complex multiplication
The elliptic curves in class 490245o do not have complex multiplication.Modular form 490245.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.