Properties

Label 36414bi
Number of curves $6$
Conductor $36414$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("36414.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 36414bi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
36414.k5 36414bi1 [1, -1, 0, -10458, -906444] [2] 163840 \(\Gamma_0(N)\)-optimal
36414.k4 36414bi2 [1, -1, 0, -218538, -39234780] [2, 2] 327680  
36414.k3 36414bi3 [1, -1, 0, -270558, -19103040] [2, 2] 655360  
36414.k1 36414bi4 [1, -1, 0, -3495798, -2514876984] [2] 655360  
36414.k6 36414bi5 [1, -1, 0, 1003932, -148336326] [2] 1310720  
36414.k2 36414bi6 [1, -1, 0, -2377368, 1397937366] [2] 1310720  

Rank

sage: E.rank()
 

The elliptic curves in class 36414bi have rank \(1\).

Modular form 36414.2.a.k

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.