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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 | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4032.bk4 | 4032bj1 | \([0, 0, 0, 36, 432]\) | \(432/7\) | \(-83607552\) | \([2]\) | \(1024\) | \(0.20070\) | \(\Gamma_0(N)\)-optimal |
| 4032.bk3 | 4032bj2 | \([0, 0, 0, -684, 6480]\) | \(740772/49\) | \(2341011456\) | \([2, 2]\) | \(2048\) | \(0.54727\) | |
| 4032.bk2 | 4032bj3 | \([0, 0, 0, -2124, -29808]\) | \(11090466/2401\) | \(229419122688\) | \([2]\) | \(4096\) | \(0.89385\) | |
| 4032.bk1 | 4032bj4 | \([0, 0, 0, -10764, 429840]\) | \(1443468546/7\) | \(668860416\) | \([2]\) | \(4096\) | \(0.89385\) |
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.
