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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 277725.cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277725.cn1 | 277725cn4 | \([1, 0, 1, -66799751, -210142994227]\) | \(14251520160844849/264449745\) | \(611688329639036015625\) | \([2]\) | \(24330240\) | \(3.1135\) | |
| 277725.cn2 | 277725cn2 | \([1, 0, 1, -4311626, -3057347977]\) | \(3832302404449/472410225\) | \(1092713556727578515625\) | \([2, 2]\) | \(12165120\) | \(2.7669\) | |
| 277725.cn3 | 277725cn1 | \([1, 0, 1, -1071501, 377184523]\) | \(58818484369/7455105\) | \(17244110879114765625\) | \([2]\) | \(6082560\) | \(2.4203\) | \(\Gamma_0(N)\)-optimal |
| 277725.cn4 | 277725cn3 | \([1, 0, 1, 6334499, -15768821227]\) | \(12152722588271/53476250625\) | \(-123693817213416884765625\) | \([2]\) | \(24330240\) | \(3.1135\) |
Rank
sage: E.rank()
The elliptic curves in class 277725.cn have rank \(1\).
Complex multiplication
The elliptic curves in class 277725.cn do not have complex multiplication.Modular form 277725.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
