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SageMath
E = EllipticCurve("gx1")
E.isogeny_class()
Elliptic curves in class 476850gx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
476850.gx4 | 476850gx1 | \([1, 1, 1, -48165613, 96774410531]\) | \(32765849647039657/8229948198912\) | \(3103920976838501952000000\) | \([2]\) | \(99090432\) | \(3.4095\) | \(\Gamma_0(N)\)-optimal* |
476850.gx2 | 476850gx2 | \([1, 1, 1, -716333613, 7378469274531]\) | \(107784459654566688937/10704361149504\) | \(4037144622610501809000000\) | \([2, 2]\) | \(198180864\) | \(3.7561\) | \(\Gamma_0(N)\)-optimal* |
476850.gx1 | 476850gx3 | \([1, 1, 1, -11461064613, 472260000720531]\) | \(441453577446719855661097/4354701912\) | \(1642373716802061375000\) | \([2]\) | \(396361728\) | \(4.1027\) | \(\Gamma_0(N)\)-optimal* |
476850.gx3 | 476850gx4 | \([1, 1, 1, -662290613, 8538772484531]\) | \(-85183593440646799657/34223681512621656\) | \(-12907444905389524884285375000\) | \([2]\) | \(396361728\) | \(4.1027\) |
Rank
sage: E.rank()
The elliptic curves in class 476850gx have rank \(2\).
Complex multiplication
The elliptic curves in class 476850gx do not have complex multiplication.Modular form 476850.2.a.gx
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.