Properties

Label 36414.j
Number of curves $6$
Conductor $36414$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("36414.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 36414.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
36414.j1 36414t6 [1, -1, 0, -35682089058, 2594328654970476] [2] 70778880  
36414.j2 36414t4 [1, -1, 0, -2234373498, 40374817041780] [2, 2] 35389440  
36414.j3 36414t5 [1, -1, 0, -761063058, 92822606071164] [2] 70778880  
36414.j4 36414t2 [1, -1, 0, -235973178, -350583079500] [2, 2] 17694720  
36414.j5 36414t1 [1, -1, 0, -182704698, -949310141004] [2] 8847360 \(\Gamma_0(N)\)-optimal
36414.j6 36414t3 [1, -1, 0, 910131462, -2758090486284] [2] 35389440  

Rank

sage: E.rank()
 

The elliptic curves in class 36414.j have rank \(0\).

Modular form 36414.2.a.j

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} - 2q^{5} - q^{7} - q^{8} + 2q^{10} + 4q^{11} - 2q^{13} + q^{14} + q^{16} + 4q^{19} + O(q^{20}) \)

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.