Properties

Label 361998.cn
Number of curves $6$
Conductor $361998$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("361998.cn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 361998.cn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
361998.cn1 361998cn5 [1, -1, 1, -20865996716, 1160135576072295] [2] 566231040  
361998.cn2 361998cn3 [1, -1, 1, -1306605956, 18054925839591] [2, 2] 283115520  
361998.cn3 361998cn6 [1, -1, 1, -445050716, 41508527204967] [2] 566231040  
361998.cn4 361998cn2 [1, -1, 1, -137991236, -156765956889] [2, 2] 141557760  
361998.cn5 361998cn1 [1, -1, 1, -106841156, -424507124505] [2] 70778880 \(\Gamma_0(N)\)-optimal
361998.cn6 361998cn4 [1, -1, 1, 532222204, -1233396826905] [2] 283115520  

Rank

sage: E.rank()
 

The elliptic curves in class 361998.cn have rank \(1\).

Modular form 361998.2.a.cn

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - 2q^{5} - q^{7} + q^{8} - 2q^{10} + 4q^{11} - q^{14} + q^{16} - q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.