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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 333270.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.bg1 | 333270bg1 | \([1, -1, 0, -2710695, -1713396739]\) | \(551105805571803/1376829440\) | \(5503144594097848320\) | \([2]\) | \(14192640\) | \(2.4734\) | \(\Gamma_0(N)\)-optimal |
333270.bg2 | 333270bg2 | \([1, -1, 0, -1695015, -3013670275]\) | \(-134745327251163/903920796800\) | \(-3612943405976661590400\) | \([2]\) | \(28385280\) | \(2.8199\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.bg do not have complex multiplication.Modular form 333270.2.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.