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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 289674.by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
289674.by1 | 289674by3 | \([1, -1, 0, -19767006, 33831608850]\) | \(661397832743623417/443352042\) | \(572574961233742698\) | \([2]\) | \(13107200\) | \(2.7236\) | |
289674.by2 | 289674by2 | \([1, -1, 0, -1243116, 521949852]\) | \(164503536215257/4178071044\) | \(5395844925532389636\) | \([2, 2]\) | \(6553600\) | \(2.3770\) | |
289674.by3 | 289674by1 | \([1, -1, 0, -175896, -16569360]\) | \(466025146777/177366672\) | \(229063375656129168\) | \([2]\) | \(3276800\) | \(2.0304\) | \(\Gamma_0(N)\)-optimal |
289674.by4 | 289674by4 | \([1, -1, 0, 205254, 1664713782]\) | \(740480746823/927484650666\) | \(-1197816718074293517354\) | \([2]\) | \(13107200\) | \(2.7236\) |
Rank
sage: E.rank()
The elliptic curves in class 289674.by have rank \(0\).
Complex multiplication
The elliptic curves in class 289674.by do not have complex multiplication.Modular form 289674.2.a.by
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.