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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 20160eu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20160.dw3 | 20160eu1 | \([0, 0, 0, -327, 1136]\) | \(82881856/36015\) | \(1680315840\) | \([2]\) | \(12288\) | \(0.46632\) | \(\Gamma_0(N)\)-optimal |
20160.dw2 | 20160eu2 | \([0, 0, 0, -2532, -48256]\) | \(601211584/11025\) | \(32920473600\) | \([2, 2]\) | \(24576\) | \(0.81289\) | |
20160.dw1 | 20160eu3 | \([0, 0, 0, -40332, -3117616]\) | \(303735479048/105\) | \(2508226560\) | \([2]\) | \(49152\) | \(1.1595\) | |
20160.dw4 | 20160eu4 | \([0, 0, 0, -12, -139984]\) | \(-8/354375\) | \(-8465264640000\) | \([2]\) | \(49152\) | \(1.1595\) |
Rank
sage: E.rank()
The elliptic curves in class 20160eu have rank \(0\).
Complex multiplication
The elliptic curves in class 20160eu do not have complex multiplication.Modular form 20160.2.a.eu
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.