Properties

Label 316239.n
Number of curves 6
Conductor 316239
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("316239.n1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 316239.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
316239.n1 316239n6 [1, 1, 0, -6186539, -5925275838] [2] 8110080  
316239.n2 316239n4 [1, 1, 0, -388824, -91615005] [2, 2] 4055040  
316239.n3 316239n2 [1, 1, 0, -53419, 2633800] [2, 2] 2027520  
316239.n4 316239n1 [1, 1, 0, -46574, 3848103] [2] 1013760 \(\Gamma_0(N)\)-optimal
316239.n5 316239n5 [1, 1, 0, 42411, -283169592] [2] 8110080  
316239.n6 316239n3 [1, 1, 0, 172466, 19304113] [2] 4055040  

Rank

sage: E.rank()
 

The elliptic curves in class 316239.n have rank \(2\).

Modular form 316239.2.a.n

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} + q^{7} - 3q^{8} + q^{9} + 2q^{10} - q^{11} + q^{12} - 6q^{13} + q^{14} - 2q^{15} - q^{16} - 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.