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SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 20160.dj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20160.dj1 | 20160bw4 | \([0, 0, 0, -508332, 139498256]\) | \(608119035935048/826875\) | \(19752284160000\) | \([4]\) | \(98304\) | \(1.8254\) | |
20160.dj2 | 20160bw3 | \([0, 0, 0, -80652, -5889616]\) | \(2428799546888/778248135\) | \(18590691833118720\) | \([2]\) | \(98304\) | \(1.8254\) | |
20160.dj3 | 20160bw2 | \([0, 0, 0, -32052, 2139104]\) | \(1219555693504/43758225\) | \(130661359718400\) | \([2, 2]\) | \(49152\) | \(1.4789\) | |
20160.dj4 | 20160bw1 | \([0, 0, 0, 753, 118316]\) | \(1012048064/130203045\) | \(-6074753267520\) | \([2]\) | \(24576\) | \(1.1323\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 20160.dj have rank \(1\).
Complex multiplication
The elliptic curves in class 20160.dj do not have complex multiplication.Modular form 20160.2.a.dj
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.