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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 42432bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42432.ba4 | 42432bk1 | \([0, -1, 0, 543, -9087]\) | \(67419143/169728\) | \(-44493176832\) | \([2]\) | \(24576\) | \(0.72702\) | \(\Gamma_0(N)\)-optimal |
| 42432.ba3 | 42432bk2 | \([0, -1, 0, -4577, -98175]\) | \(40459583737/7033104\) | \(1843686014976\) | \([2, 2]\) | \(49152\) | \(1.0736\) | |
| 42432.ba2 | 42432bk3 | \([0, -1, 0, -21217, 1103233]\) | \(4029546653497/351790452\) | \(92219756249088\) | \([2]\) | \(98304\) | \(1.4202\) | |
| 42432.ba1 | 42432bk4 | \([0, -1, 0, -69857, -7083135]\) | \(143820170742457/5826444\) | \(1527367335936\) | \([2]\) | \(98304\) | \(1.4202\) |
Rank
sage: E.rank()
The elliptic curves in class 42432bk have rank \(1\).
Complex multiplication
The elliptic curves in class 42432bk do not have complex multiplication.Modular form 42432.2.a.bk
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.
