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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 388416bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388416.bh3 | 388416bh1 | \([0, -1, 0, -137949, -19674435]\) | \(11745974272/357\) | \(8823922824192\) | \([2]\) | \(1474560\) | \(1.5819\) | \(\Gamma_0(N)\)-optimal |
388416.bh2 | 388416bh2 | \([0, -1, 0, -143729, -17930031]\) | \(830321872/127449\) | \(50402247171784704\) | \([2, 2]\) | \(2949120\) | \(1.9285\) | |
388416.bh1 | 388416bh3 | \([0, -1, 0, -629249, 174821409]\) | \(17418812548/1753941\) | \(2774523701456338944\) | \([2]\) | \(5898240\) | \(2.2751\) | |
388416.bh4 | 388416bh4 | \([0, -1, 0, 249311, -99132095]\) | \(1083360092/3306177\) | \(-5229974353589895168\) | \([2]\) | \(5898240\) | \(2.2751\) |
Rank
sage: E.rank()
The elliptic curves in class 388416bh have rank \(1\).
Complex multiplication
The elliptic curves in class 388416bh do not have complex multiplication.Modular form 388416.2.a.bh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.