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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 27200cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27200.g2 | 27200cn1 | \([0, 1, 0, -633, -6137]\) | \(438976/17\) | \(1088000000\) | \([2]\) | \(15360\) | \(0.50150\) | \(\Gamma_0(N)\)-optimal |
27200.g1 | 27200cn2 | \([0, 1, 0, -1633, 16863]\) | \(941192/289\) | \(147968000000\) | \([2]\) | \(30720\) | \(0.84808\) |
Rank
sage: E.rank()
The elliptic curves in class 27200cn have rank \(1\).
Complex multiplication
The elliptic curves in class 27200cn do not have complex multiplication.Modular form 27200.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.