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SageMath
E = EllipticCurve("fp1")
E.isogeny_class()
Elliptic curves in class 476850.fp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
476850.fp1 | 476850fp4 | \([1, 0, 1, -10145684726, -393342696764152]\) | \(306234591284035366263793/1727485056\) | \(651520152119826000000\) | \([2]\) | \(396361728\) | \(4.0636\) | |
476850.fp2 | 476850fp2 | \([1, 0, 1, -634116726, -6145786620152]\) | \(74768347616680342513/5615307472896\) | \(2117810493488169216000000\) | \([2, 2]\) | \(198180864\) | \(3.7170\) | |
476850.fp3 | 476850fp3 | \([1, 0, 1, -592500726, -6987262140152]\) | \(-60992553706117024753/20624795251201152\) | \(-7778631538855314676242000000\) | \([2]\) | \(396361728\) | \(4.0636\) | |
476850.fp4 | 476850fp1 | \([1, 0, 1, -42244726, -82649852152]\) | \(22106889268753393/4969545596928\) | \(1874261714757746688000000\) | \([2]\) | \(99090432\) | \(3.3704\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 476850.fp have rank \(1\).
Complex multiplication
The elliptic curves in class 476850.fp do not have complex multiplication.Modular form 476850.2.a.fp
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.