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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 222768cl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
222768.fz1 | 222768cl1 | \([0, 0, 0, -65523, 6452210]\) | \(10418796526321/6390657\) | \(19082399551488\) | \([2]\) | \(1146880\) | \(1.4911\) | \(\Gamma_0(N)\)-optimal |
222768.fz2 | 222768cl2 | \([0, 0, 0, -53283, 8936930]\) | \(-5602762882081/8312741073\) | \(-24821711840120832\) | \([2]\) | \(2293760\) | \(1.8377\) |
Rank
sage: E.rank()
The elliptic curves in class 222768cl have rank \(0\).
Complex multiplication
The elliptic curves in class 222768cl do not have complex multiplication.Modular form 222768.2.a.cl
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.