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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 216384cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
216384.k2 | 216384cn1 | \([0, -1, 0, 9343, 1533729]\) | \(2924207/34776\) | \(-1072525901561856\) | \([]\) | \(1105920\) | \(1.5650\) | \(\Gamma_0(N)\)-optimal |
216384.k1 | 216384cn2 | \([0, -1, 0, -84737, -43304799]\) | \(-2181825073/25039686\) | \(-772248441510690816\) | \([]\) | \(3317760\) | \(2.1143\) |
Rank
sage: E.rank()
The elliptic curves in class 216384cn have rank \(0\).
Complex multiplication
The elliptic curves in class 216384cn do not have complex multiplication.Modular form 216384.2.a.cn
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.