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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 77616.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 77616.bg1 | 77616bu4 | \([0, 0, 0, -310611, -66630494]\) | \(37736227588/33\) | \(2898208760832\) | \([2]\) | \(442368\) | \(1.6916\) | |
| 77616.bg2 | 77616bu3 | \([0, 0, 0, -46011, 2345434]\) | \(122657188/43923\) | \(3857515860667392\) | \([2]\) | \(442368\) | \(1.6916\) | |
| 77616.bg3 | 77616bu2 | \([0, 0, 0, -19551, -1025570]\) | \(37642192/1089\) | \(23910222276864\) | \([2, 2]\) | \(221184\) | \(1.3450\) | |
| 77616.bg4 | 77616bu1 | \([0, 0, 0, 294, -53165]\) | \(2048/891\) | \(-1222681820976\) | \([2]\) | \(110592\) | \(0.99844\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 77616.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 77616.bg do not have complex multiplication.Modular form 77616.2.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 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.
