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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 225264cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
225264.v1 | 225264cp1 | \([0, -1, 0, -1203, -6966]\) | \(256000/117\) | \(88069889232\) | \([2]\) | \(221184\) | \(0.79504\) | \(\Gamma_0(N)\)-optimal |
225264.v2 | 225264cp2 | \([0, -1, 0, 4212, -56784]\) | \(686000/507\) | \(-6106178986752\) | \([2]\) | \(442368\) | \(1.1416\) |
Rank
sage: E.rank()
The elliptic curves in class 225264cp have rank \(1\).
Complex multiplication
The elliptic curves in class 225264cp do not have complex multiplication.Modular form 225264.2.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.