Properties

Label 36822.r
Number of curves $4$
Conductor $36822$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 36822.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
36822.r1 36822p3 \([1, 1, 1, -270938, 40925927]\) \(46753267515625/11591221248\) \(545319215478079488\) \([2]\) \(497664\) \(2.1136\)  
36822.r2 36822p1 \([1, 1, 1, -92243, -10817647]\) \(1845026709625/793152\) \(37314534606912\) \([2]\) \(165888\) \(1.5643\) \(\Gamma_0(N)\)-optimal
36822.r3 36822p2 \([1, 1, 1, -77803, -14300575]\) \(-1107111813625/1228691592\) \(-57804878422932552\) \([2]\) \(331776\) \(1.9109\)  
36822.r4 36822p4 \([1, 1, 1, 653222, 260506343]\) \(655215969476375/1001033261568\) \(-47094491700770001408\) \([2]\) \(995328\) \(2.4602\)  

Rank

sage: E.rank()
 

The elliptic curves in class 36822.r have rank \(1\).

Complex multiplication

The elliptic curves in class 36822.r do not have complex multiplication.

Modular form 36822.2.a.r

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + 2 q^{7} + q^{8} + q^{9} - q^{12} - 2 q^{13} + 2 q^{14} + q^{16} - q^{17} + q^{18} + O(q^{20})\) 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 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.