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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 255024cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
255024.cp2 | 255024cp1 | \([0, 0, 0, -558435, 160806242]\) | \(-6449916994998625/8532911772\) | \(-25479138024603648\) | \([2]\) | \(2064384\) | \(2.0545\) | \(\Gamma_0(N)\)-optimal |
255024.cp1 | 255024cp2 | \([0, 0, 0, -8937795, 10284748994]\) | \(26444015547214434625/46191222\) | \(137926249832448\) | \([2]\) | \(4128768\) | \(2.4010\) |
Rank
sage: E.rank()
The elliptic curves in class 255024cp have rank \(1\).
Complex multiplication
The elliptic curves in class 255024cp do not have complex multiplication.Modular form 255024.2.a.cp
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.