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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 485100cs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.cs3 | 485100cs1 | \([0, 0, 0, -1264200, 44204125]\) | \(281370820608/161767375\) | \(128464446834281250000\) | \([2]\) | \(11943936\) | \(2.5481\) | \(\Gamma_0(N)\)-optimal* |
485100.cs4 | 485100cs2 | \([0, 0, 0, 5038425, 353032750]\) | \(1113258734352/648484375\) | \(-8239702129312500000000\) | \([2]\) | \(23887872\) | \(2.8947\) | |
485100.cs1 | 485100cs3 | \([0, 0, 0, -73294200, 241518776625]\) | \(75216478666752/326095\) | \(188783346785591250000\) | \([2]\) | \(35831808\) | \(3.0974\) | \(\Gamma_0(N)\)-optimal* |
485100.cs2 | 485100cs4 | \([0, 0, 0, -72136575, 249516807750]\) | \(-4481782160112/310023175\) | \(-2871664395104250900000000\) | \([2]\) | \(71663616\) | \(3.4440\) |
Rank
sage: E.rank()
The elliptic curves in class 485100cs have rank \(1\).
Complex multiplication
The elliptic curves in class 485100cs do not have complex multiplication.Modular form 485100.2.a.cs
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.