Properties

Label 209814y
Number of curves $4$
Conductor $209814$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("y1")
 
E.isogeny_class()
 

Elliptic curves in class 209814y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
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.cf1 209814y3 \([1, 1, 1, -26244963, 39114470913]\) \(46753267515625/11591221248\) \(495654250394187557240832\) \([2]\) \(29859840\) \(3.2570\)  
209814.cf4 209814y4 \([1, 1, 1, 63275677, 248127261185]\) \(655215969476375/1001033261568\) \(-42805359354843342223667712\) \([2]\) \(59719680\) \(3.6036\)  

Rank

sage: E.rank()
 

The elliptic curves in class 209814y have rank \(1\).

Complex multiplication

The elliptic curves in class 209814y do not have complex multiplication.

Modular form 209814.2.a.y

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + 2 q^{7} + q^{8} + q^{9} - q^{12} - 2 q^{13} + 2 q^{14} + q^{16} + q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.