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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 425880.cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
425880.cn1 | 425880cn3 | \([0, 0, 0, -37989003, 90122728358]\) | \(841356017734178/1404585\) | \(10121983396954490880\) | \([2]\) | \(27525120\) | \(2.9090\) | \(\Gamma_0(N)\)-optimal* |
425880.cn2 | 425880cn4 | \([0, 0, 0, -6230523, -4131840778]\) | \(3711757787138/1124589375\) | \(8104226502590749440000\) | \([2]\) | \(27525120\) | \(2.9090\) | |
425880.cn3 | 425880cn2 | \([0, 0, 0, -2397603, 1379131598]\) | \(423026849956/16769025\) | \(60422043747126297600\) | \([2, 2]\) | \(13762560\) | \(2.5624\) | \(\Gamma_0(N)\)-optimal* |
425880.cn4 | 425880cn1 | \([0, 0, 0, 66417, 78621842]\) | \(35969456/2985255\) | \(-2689112935998478080\) | \([2]\) | \(6881280\) | \(2.2158\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 425880.cn have rank \(0\).
Complex multiplication
The elliptic curves in class 425880.cn do not have complex multiplication.Modular form 425880.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.