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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 6762.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6762.bg1 | 6762bi3 | \([1, 0, 0, -12104, -513486]\) | \(1666957239793/301806\) | \(35507174094\) | \([2]\) | \(12288\) | \(1.0280\) | |
| 6762.bg2 | 6762bi4 | \([1, 0, 0, -5244, 140958]\) | \(135559106353/5037138\) | \(592614248562\) | \([2]\) | \(12288\) | \(1.0280\) | |
| 6762.bg3 | 6762bi2 | \([1, 0, 0, -834, -6336]\) | \(545338513/171396\) | \(20164568004\) | \([2, 2]\) | \(6144\) | \(0.68140\) | |
| 6762.bg4 | 6762bi1 | \([1, 0, 0, 146, -652]\) | \(2924207/3312\) | \(-389653488\) | \([2]\) | \(3072\) | \(0.33482\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6762.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 6762.bg do not have complex multiplication.Modular form 6762.2.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.
