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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 47040cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.fk3 | 47040cn1 | \([0, 1, 0, -44116, 3551834]\) | \(1261112198464/675\) | \(5082436800\) | \([2]\) | \(110592\) | \(1.1920\) | \(\Gamma_0(N)\)-optimal |
| 47040.fk2 | 47040cn2 | \([0, 1, 0, -44361, 3510135]\) | \(20034997696/455625\) | \(219561269760000\) | \([2, 2]\) | \(221184\) | \(1.5386\) | |
| 47040.fk4 | 47040cn3 | \([0, 1, 0, 4639, 10889535]\) | \(2863288/13286025\) | \(-51219253009612800\) | \([2]\) | \(442368\) | \(1.8852\) | |
| 47040.fk1 | 47040cn4 | \([0, 1, 0, -97281, -6534081]\) | \(26410345352/10546875\) | \(40659494400000000\) | \([2]\) | \(442368\) | \(1.8852\) |
Rank
sage: E.rank()
The elliptic curves in class 47040cn have rank \(1\).
Complex multiplication
The elliptic curves in class 47040cn do not have complex multiplication.Modular form 47040.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
