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SageMath
sage: E = EllipticCurve("bj1")
sage: E.isogeny_class()
Elliptic curves in class 4032bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
4032.bk4 | 4032bj1 | [0, 0, 0, 36, 432] | [2] | 1024 | \(\Gamma_0(N)\)-optimal |
4032.bk3 | 4032bj2 | [0, 0, 0, -684, 6480] | [2, 2] | 2048 | |
4032.bk2 | 4032bj3 | [0, 0, 0, -2124, -29808] | [2] | 4096 | |
4032.bk1 | 4032bj4 | [0, 0, 0, -10764, 429840] | [2] | 4096 |
Rank
sage: E.rank()
The elliptic curves in class 4032bj have rank \(0\).
Complex multiplication
The elliptic curves in class 4032bj do not have complex multiplication.Modular form 4032.2.a.bj
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.