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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 33600cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.fd4 | 33600cn1 | \([0, 1, 0, 667, -2037]\) | \(2048000/1323\) | \(-21168000000\) | \([2]\) | \(27648\) | \(0.67022\) | \(\Gamma_0(N)\)-optimal |
33600.fd3 | 33600cn2 | \([0, 1, 0, -2833, -19537]\) | \(9826000/5103\) | \(1306368000000\) | \([2]\) | \(55296\) | \(1.0168\) | |
33600.fd2 | 33600cn3 | \([0, 1, 0, -11333, -482037]\) | \(-10061824000/352947\) | \(-5647152000000\) | \([2]\) | \(82944\) | \(1.2195\) | |
33600.fd1 | 33600cn4 | \([0, 1, 0, -182833, -30151537]\) | \(2640279346000/3087\) | \(790272000000\) | \([2]\) | \(165888\) | \(1.5661\) |
Rank
sage: E.rank()
The elliptic curves in class 33600cn have rank \(0\).
Complex multiplication
The elliptic curves in class 33600cn 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 Cremona 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 Cremona labels.