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SageMath
sage: E = EllipticCurve("dk1")
sage: E.isogeny_class()
Elliptic curves in class 123840.dk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
123840.dk1 | 123840cd4 | [0, 0, 0, -482988, -128316112] | [2] | 1179648 | |
123840.dk2 | 123840cd2 | [0, 0, 0, -50988, 1111088] | [2, 2] | 589824 | |
123840.dk3 | 123840cd1 | [0, 0, 0, -39468, 3014192] | [2] | 294912 | \(\Gamma_0(N)\)-optimal |
123840.dk4 | 123840cd3 | [0, 0, 0, 196692, 8739632] | [2] | 1179648 |
Rank
sage: E.rank()
The elliptic curves in class 123840.dk have rank \(1\).
Complex multiplication
The elliptic curves in class 123840.dk do not have complex multiplication.Modular form 123840.2.a.dk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.