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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 209814.cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
209814.cf1 | 209814y3 | \([1, 1, 1, -26244963, 39114470913]\) | \(46753267515625/11591221248\) | \(495654250394187557240832\) | \([2]\) | \(29859840\) | \(3.2570\) | |
209814.cf2 | 209814y1 | \([1, 1, 1, -8935308, -10280360595]\) | \(1845026709625/793152\) | \(33916112167773768768\) | \([2]\) | \(9953280\) | \(2.7077\) | \(\Gamma_0(N)\)-optimal |
209814.cf3 | 209814y2 | \([1, 1, 1, -7536548, -13606052371]\) | \(-1107111813625/1228691592\) | \(-52540297261902539542728\) | \([2]\) | \(19906560\) | \(3.0542\) | |
209814.cf4 | 209814y4 | \([1, 1, 1, 63275677, 248127261185]\) | \(655215969476375/1001033261568\) | \(-42805359354843342223667712\) | \([2]\) | \(59719680\) | \(3.6036\) |
Rank
sage: E.rank()
The elliptic curves in class 209814.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 209814.cf do not have complex multiplication.Modular form 209814.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.