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SageMath
E = EllipticCurve("ee1")
E.isogeny_class()
Elliptic curves in class 20160ee
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20160.cp3 | 20160ee1 | \([0, 0, 0, -2028, -22192]\) | \(4826809/1680\) | \(321052999680\) | \([2]\) | \(24576\) | \(0.90946\) | \(\Gamma_0(N)\)-optimal |
20160.cp2 | 20160ee2 | \([0, 0, 0, -13548, 590672]\) | \(1439069689/44100\) | \(8427641241600\) | \([2, 2]\) | \(49152\) | \(1.2560\) | |
20160.cp1 | 20160ee3 | \([0, 0, 0, -215148, 38410832]\) | \(5763259856089/5670\) | \(1083553873920\) | \([2]\) | \(98304\) | \(1.6026\) | |
20160.cp4 | 20160ee4 | \([0, 0, 0, 3732, 1993808]\) | \(30080231/9003750\) | \(-1720643420160000\) | \([2]\) | \(98304\) | \(1.6026\) |
Rank
sage: E.rank()
The elliptic curves in class 20160ee have rank \(0\).
Complex multiplication
The elliptic curves in class 20160ee do not have complex multiplication.Modular form 20160.2.a.ee
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.