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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 444675bw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
444675.bw3 | 444675bw1 | \([1, 1, 1, -1633563, -641557344]\) | \(148035889/31185\) | \(101557061298054140625\) | \([2]\) | \(13271040\) | \(2.5536\) | \(\Gamma_0(N)\)-optimal* |
444675.bw2 | 444675bw2 | \([1, 1, 1, -8303688, 8643256656]\) | \(19443408769/1334025\) | \(4344385399972316015625\) | \([2, 2]\) | \(26542080\) | \(2.9002\) | \(\Gamma_0(N)\)-optimal* |
444675.bw1 | 444675bw3 | \([1, 1, 1, -130589313, 574336557906]\) | \(75627935783569/396165\) | \(1290150815749354453125\) | \([2]\) | \(53084160\) | \(3.2468\) | \(\Gamma_0(N)\)-optimal* |
444675.bw4 | 444675bw4 | \([1, 1, 1, 7259937, 37311453906]\) | \(12994449551/192163125\) | \(-625798373091250283203125\) | \([2]\) | \(53084160\) | \(3.2468\) |
Rank
sage: E.rank()
The elliptic curves in class 444675bw have rank \(1\).
Complex multiplication
The elliptic curves in class 444675bw do not have complex multiplication.Modular form 444675.2.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 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.