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SageMath
E = EllipticCurve("kg1")
E.isogeny_class()
Elliptic curves in class 486720.kg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.kg1 | 486720kg2 | \([0, 0, 0, -4479852, 3269382064]\) | \(10779215329/1232010\) | \(1136427663619420323840\) | \([2]\) | \(24772608\) | \(2.7725\) | \(\Gamma_0(N)\)-optimal* |
486720.kg2 | 486720kg1 | \([0, 0, 0, 387348, 257558704]\) | \(6967871/35100\) | \(-32376856513373798400\) | \([2]\) | \(12386304\) | \(2.4259\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 486720.kg have rank \(1\).
Complex multiplication
The elliptic curves in class 486720.kg do not have complex multiplication.Modular form 486720.2.a.kg
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.