Properties

Label 36.a
Number of curves $4$
Conductor $36$
CM -3
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("36.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 36.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
36.a1 36a4 [0, 0, 0, -135, -594] [2] 6  
36.a2 36a2 [0, 0, 0, -15, 22] [6] 2  
36.a3 36a3 [0, 0, 0, 0, -27] [2] 3  
36.a4 36a1 [0, 0, 0, 0, 1] [6] 1 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 36.a have rank \(0\).

Modular form 36.2.a.a

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

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.

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.