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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 422370d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
422370.d4 | 422370d1 | \([1, -1, 0, 35130, 1969396]\) | \(3774555693/3515200\) | \(-4465143384062400\) | \([2]\) | \(3981312\) | \(1.6901\) | \(\Gamma_0(N)\)-optimal* |
422370.d3 | 422370d2 | \([1, -1, 0, -181470, 17867836]\) | \(520300455507/193072360\) | \(245248000369627320\) | \([2]\) | \(7962624\) | \(2.0367\) | \(\Gamma_0(N)\)-optimal* |
422370.d2 | 422370d3 | \([1, -1, 0, -809610, 283680800]\) | \(-63378025803/812500\) | \(-752378311524937500\) | \([2]\) | \(11943936\) | \(2.2395\) | \(\Gamma_0(N)\)-optimal* |
422370.d1 | 422370d4 | \([1, -1, 0, -12993360, 18030531050]\) | \(261984288445803/42250\) | \(39123672199296750\) | \([2]\) | \(23887872\) | \(2.5860\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 422370d have rank \(0\).
Complex multiplication
The elliptic curves in class 422370d do not have complex multiplication.Modular form 422370.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.