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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 433200gk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.gk2 | 433200gk1 | \([0, 1, 0, -11338408, -51548084812]\) | \(-53540005609/350208000\) | \(-1054454009167872000000000\) | \([2]\) | \(69672960\) | \(3.2930\) | \(\Gamma_0(N)\)-optimal* |
433200.gk1 | 433200gk2 | \([0, 1, 0, -288586408, -1883048372812]\) | \(882774443450089/2166000000\) | \(6521688207744000000000000\) | \([2]\) | \(139345920\) | \(3.6396\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 433200gk have rank \(0\).
Complex multiplication
The elliptic curves in class 433200gk do not have complex multiplication.Modular form 433200.2.a.gk
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.