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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 33600.cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.cn1 | 33600fg4 | \([0, -1, 0, -182833, 30151537]\) | \(2640279346000/3087\) | \(790272000000\) | \([2]\) | \(165888\) | \(1.5661\) | |
33600.cn2 | 33600fg3 | \([0, -1, 0, -11333, 482037]\) | \(-10061824000/352947\) | \(-5647152000000\) | \([2]\) | \(82944\) | \(1.2195\) | |
33600.cn3 | 33600fg2 | \([0, -1, 0, -2833, 19537]\) | \(9826000/5103\) | \(1306368000000\) | \([2]\) | \(55296\) | \(1.0168\) | |
33600.cn4 | 33600fg1 | \([0, -1, 0, 667, 2037]\) | \(2048000/1323\) | \(-21168000000\) | \([2]\) | \(27648\) | \(0.67022\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33600.cn have rank \(1\).
Complex multiplication
The elliptic curves in class 33600.cn do not have complex multiplication.Modular form 33600.2.a.cn
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.