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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 51520bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51520.ck3 | 51520bg1 | \([0, -1, 0, -9665, 358337]\) | \(380920459249/12622400\) | \(3308886425600\) | \([2]\) | \(110592\) | \(1.1753\) | \(\Gamma_0(N)\)-optimal |
| 51520.ck4 | 51520bg2 | \([0, -1, 0, 3135, 1226177]\) | \(12994449551/2489452840\) | \(-652595125288960\) | \([2]\) | \(221184\) | \(1.5219\) | |
| 51520.ck1 | 51520bg3 | \([0, -1, 0, -108225, -13549375]\) | \(534774372149809/5323062500\) | \(1395408896000000\) | \([2]\) | \(331776\) | \(1.7246\) | |
| 51520.ck2 | 51520bg4 | \([0, -1, 0, -28225, -33213375]\) | \(-9486391169809/1813439640250\) | \(-475382321053696000\) | \([2]\) | \(663552\) | \(2.0712\) |
Rank
sage: E.rank()
The elliptic curves in class 51520bg have rank \(1\).
Complex multiplication
The elliptic curves in class 51520bg do not have complex multiplication.Modular form 51520.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
