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SageMath
sage: E = EllipticCurve("bb1")
sage: E.isogeny_class()
Elliptic curves in class 4032.bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4032.bb1 | 4032g3 | \([0, 0, 0, -10764, -429840]\) | \(1443468546/7\) | \(668860416\) | \([2]\) | \(4096\) | \(0.89385\) | |
| 4032.bb2 | 4032g4 | \([0, 0, 0, -2124, 29808]\) | \(11090466/2401\) | \(229419122688\) | \([2]\) | \(4096\) | \(0.89385\) | |
| 4032.bb3 | 4032g2 | \([0, 0, 0, -684, -6480]\) | \(740772/49\) | \(2341011456\) | \([2, 2]\) | \(2048\) | \(0.54727\) | |
| 4032.bb4 | 4032g1 | \([0, 0, 0, 36, -432]\) | \(432/7\) | \(-83607552\) | \([2]\) | \(1024\) | \(0.20070\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4032.bb have rank \(0\).
Complex multiplication
The elliptic curves in class 4032.bb do not have complex multiplication.Modular form 4032.2.a.bb
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
