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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 4050.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4050.bg1 | 4050be2 | \([1, -1, 1, -14555, -675053]\) | \(-1642545/8\) | \(-1660753125000\) | \([]\) | \(9720\) | \(1.1934\) | |
4050.bg2 | 4050be1 | \([1, -1, 1, 445, -5053]\) | \(308655/512\) | \(-16200000000\) | \([3]\) | \(3240\) | \(0.64408\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4050.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 4050.bg do not have complex multiplication.Modular form 4050.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.