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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 400710bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
400710.bg3 | 400710bg1 | \([1, 1, 1, -112820, -17978443]\) | \(-3375675045001/999000000\) | \(-46998835119000000\) | \([2]\) | \(7185024\) | \(1.9135\) | \(\Gamma_0(N)\)-optimal* |
400710.bg2 | 400710bg2 | \([1, 1, 1, -1917820, -1023002443]\) | \(16581570075765001/998001000\) | \(46951836283881000\) | \([2]\) | \(14370048\) | \(2.2601\) | \(\Gamma_0(N)\)-optimal* |
400710.bg4 | 400710bg3 | \([1, 1, 1, 834805, 149561657]\) | \(1367594037332999/995878502400\) | \(-46851981514368614400\) | \([2]\) | \(21555072\) | \(2.4628\) | \(\Gamma_0(N)\)-optimal* |
400710.bg1 | 400710bg4 | \([1, 1, 1, -3785995, 1264098617]\) | \(127568139540190201/59114336463360\) | \(2781086038649195420160\) | \([2]\) | \(43110144\) | \(2.8094\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 400710bg have rank \(0\).
Complex multiplication
The elliptic curves in class 400710bg do not have complex multiplication.Modular form 400710.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.