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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 5040.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.bg1 | 5040bo3 | \([0, 0, 0, -53787, 4801354]\) | \(5763259856089/5670\) | \(16930529280\) | \([2]\) | \(12288\) | \(1.2560\) | |
5040.bg2 | 5040bo2 | \([0, 0, 0, -3387, 73834]\) | \(1439069689/44100\) | \(131681894400\) | \([2, 2]\) | \(6144\) | \(0.90946\) | |
5040.bg3 | 5040bo1 | \([0, 0, 0, -507, -2774]\) | \(4826809/1680\) | \(5016453120\) | \([2]\) | \(3072\) | \(0.56289\) | \(\Gamma_0(N)\)-optimal |
5040.bg4 | 5040bo4 | \([0, 0, 0, 933, 249226]\) | \(30080231/9003750\) | \(-26885053440000\) | \([4]\) | \(12288\) | \(1.2560\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.bg do not have complex multiplication.Modular form 5040.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.