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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 66654.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66654.l1 | 66654v4 | \([1, -1, 0, -851500188, -9563471523270]\) | \(632678989847546725777/80515134\) | \(8689045361500567854\) | \([2]\) | \(16220160\) | \(3.4940\) | |
66654.l2 | 66654v3 | \([1, -1, 0, -60888528, -103543708266]\) | \(231331938231569617/90942310746882\) | \(9814327122144990588263442\) | \([2]\) | \(16220160\) | \(3.4940\) | |
66654.l3 | 66654v2 | \([1, -1, 0, -53223318, -149392395360]\) | \(154502321244119857/55101928644\) | \(5946498921480817192164\) | \([2, 2]\) | \(8110080\) | \(3.1475\) | |
66654.l4 | 66654v1 | \([1, -1, 0, -2851938, -3023239356]\) | \(-23771111713777/22848457968\) | \(-2465763613139997879408\) | \([2]\) | \(4055040\) | \(2.8009\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 66654.l have rank \(0\).
Complex multiplication
The elliptic curves in class 66654.l do not have complex multiplication.Modular form 66654.2.a.l
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.