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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 350658.ed
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
350658.ed1 | 350658ed4 | \([1, -1, 1, -194766584, -1046163516307]\) | \(632678989847546725777/80515134\) | \(103982716580742846\) | \([2]\) | \(39321600\) | \(3.1252\) | |
350658.ed2 | 350658ed3 | \([1, -1, 1, -13927244, -11325271075]\) | \(231331938231569617/90942310746882\) | \(117449081356442569622658\) | \([2]\) | \(39321600\) | \(3.1252\) | |
350658.ed3 | 350658ed2 | \([1, -1, 1, -12173954, -16341083107]\) | \(154502321244119857/55101928644\) | \(71162375873849604036\) | \([2, 2]\) | \(19660800\) | \(2.7787\) | |
350658.ed4 | 350658ed1 | \([1, -1, 1, -652334, -330639955]\) | \(-23771111713777/22848457968\) | \(-29508051606714826992\) | \([2]\) | \(9830400\) | \(2.4321\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 350658.ed have rank \(1\).
Complex multiplication
The elliptic curves in class 350658.ed do not have complex multiplication.Modular form 350658.2.a.ed
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.