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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 4032.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4032.u1 | 4032n4 | \([0, 0, 0, -65820, 6499568]\) | \(2640279346000/3087\) | \(36870930432\) | \([2]\) | \(9216\) | \(1.3107\) | |
| 4032.u2 | 4032n3 | \([0, 0, 0, -4080, 103304]\) | \(-10061824000/352947\) | \(-263473523712\) | \([2]\) | \(4608\) | \(0.96412\) | |
| 4032.u3 | 4032n2 | \([0, 0, 0, -1020, 4016]\) | \(9826000/5103\) | \(60949905408\) | \([2]\) | \(3072\) | \(0.76138\) | |
| 4032.u4 | 4032n1 | \([0, 0, 0, 240, 488]\) | \(2048000/1323\) | \(-987614208\) | \([2]\) | \(1536\) | \(0.41481\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4032.u have rank \(1\).
Complex multiplication
The elliptic curves in class 4032.u do not have complex multiplication.Modular form 4032.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
