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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 270504bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 270504.bg1 | 270504bg1 | \([0, 0, 0, -8670, 142477]\) | \(256000/117\) | \(32940250763472\) | \([2]\) | \(589824\) | \(1.2887\) | \(\Gamma_0(N)\)-optimal |
| 270504.bg2 | 270504bg2 | \([0, 0, 0, 30345, 1071034]\) | \(686000/507\) | \(-2283857386267392\) | \([2]\) | \(1179648\) | \(1.6353\) |
Rank
sage: E.rank()
The elliptic curves in class 270504bg have rank \(1\).
Complex multiplication
The elliptic curves in class 270504bg do not have complex multiplication.Modular form 270504.2.a.bg
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
