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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 7200.bw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7200.bw1 | 7200bo2 | \([0, 0, 0, -810300, -280748000]\) | \(1261112198464/675\) | \(31492800000000\) | \([2]\) | \(73728\) | \(1.9197\) | |
7200.bw2 | 7200bo3 | \([0, 0, 0, -111675, 7996750]\) | \(26410345352/10546875\) | \(61509375000000000\) | \([2]\) | \(73728\) | \(1.9197\) | |
7200.bw3 | 7200bo1 | \([0, 0, 0, -50925, -4335500]\) | \(20034997696/455625\) | \(332150625000000\) | \([2, 2]\) | \(36864\) | \(1.5731\) | \(\Gamma_0(N)\)-optimal |
7200.bw4 | 7200bo4 | \([0, 0, 0, 5325, -13391750]\) | \(2863288/13286025\) | \(-77484097800000000\) | \([2]\) | \(73728\) | \(1.9197\) |
Rank
sage: E.rank()
The elliptic curves in class 7200.bw have rank \(1\).
Complex multiplication
The elliptic curves in class 7200.bw do not have complex multiplication.Modular form 7200.2.a.bw
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.