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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 43350cy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43350.dl2 | 43350cy1 | \([1, 0, 0, -1846138, 964969892]\) | \(1845026709625/793152\) | \(299136892617000000\) | \([2]\) | \(995328\) | \(2.3134\) | \(\Gamma_0(N)\)-optimal |
43350.dl3 | 43350cy2 | \([1, 0, 0, -1557138, 1277378892]\) | \(-1107111813625/1228691592\) | \(-463400438775310125000\) | \([2]\) | \(1990656\) | \(2.6600\) | |
43350.dl1 | 43350cy3 | \([1, 0, 0, -5422513, -3674888983]\) | \(46753267515625/11591221248\) | \(4371623479185408000000\) | \([2]\) | \(2985984\) | \(2.8628\) | |
43350.dl4 | 43350cy4 | \([1, 0, 0, 13073487, -23299144983]\) | \(655215969476375/1001033261568\) | \(-377539209724885128000000\) | \([2]\) | \(5971968\) | \(3.2093\) |
Rank
sage: E.rank()
The elliptic curves in class 43350cy have rank \(1\).
Complex multiplication
The elliptic curves in class 43350cy do not have complex multiplication.Modular form 43350.2.a.cy
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.