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SageMath
E = EllipticCurve("ee1")
E.isogeny_class()
Elliptic curves in class 259920ee
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259920.ee2 | 259920ee1 | \([0, 0, 0, -5121507, -4894760734]\) | \(-105756712489/12476160\) | \(-1752629097718103408640\) | \([2]\) | \(13271040\) | \(2.8121\) | \(\Gamma_0(N)\)-optimal |
259920.ee1 | 259920ee2 | \([0, 0, 0, -84137187, -297047335966]\) | \(468898230633769/5540400\) | \(778305684841920921600\) | \([2]\) | \(26542080\) | \(3.1587\) |
Rank
sage: E.rank()
The elliptic curves in class 259920ee have rank \(1\).
Complex multiplication
The elliptic curves in class 259920ee do not have complex multiplication.Modular form 259920.2.a.ee
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.