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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 102850.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
102850.bg1 | 102850bi2 | \([1, -1, 0, -3633515617, -84301221843459]\) | \(1151968490735775903/342102016\) | \(1575505192800088000000000\) | \([2]\) | \(62853120\) | \(4.0081\) | |
102850.bg2 | 102850bi1 | \([1, -1, 0, -226155617, -1328598483459]\) | \(-277767636824223/4848615424\) | \(-22329651452280832000000000\) | \([2]\) | \(31426560\) | \(3.6616\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 102850.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 102850.bg do not have complex multiplication.Modular form 102850.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.