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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 43350cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43350.ci4 | 43350cn1 | \([1, 1, 1, -873, -13869]\) | \(-24389/12\) | \(-36206353500\) | \([2]\) | \(40960\) | \(0.73134\) | \(\Gamma_0(N)\)-optimal |
43350.ci2 | 43350cn2 | \([1, 1, 1, -15323, -736369]\) | \(131872229/18\) | \(54309530250\) | \([2]\) | \(81920\) | \(1.0779\) | |
43350.ci3 | 43350cn3 | \([1, 1, 1, -8098, 1344431]\) | \(-19465109/248832\) | \(-750774946176000\) | \([2]\) | \(204800\) | \(1.5361\) | |
43350.ci1 | 43350cn4 | \([1, 1, 1, -239298, 44810031]\) | \(502270291349/1889568\) | \(5701197247524000\) | \([2]\) | \(409600\) | \(1.8826\) |
Rank
sage: E.rank()
The elliptic curves in class 43350cn have rank \(1\).
Complex multiplication
The elliptic curves in class 43350cn do not have complex multiplication.Modular form 43350.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.