Properties

Label 32368a
Number of curves $4$
Conductor $32368$
CM no
Rank $1$
Graph

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

Elliptic curves in class 32368a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
32368.q4 32368a1 [0, 0, 0, 289, -9826] [2] 18432 \(\Gamma_0(N)\)-optimal
32368.q3 32368a2 [0, 0, 0, -5491, -147390] [2, 2] 36864  
32368.q2 32368a3 [0, 0, 0, -17051, 677994] [2] 73728  
32368.q1 32368a4 [0, 0, 0, -86411, -9776870] [2] 73728  

Rank

sage: E.rank()
 

The elliptic curves in class 32368a have rank \(1\).

Complex multiplication

The elliptic curves in class 32368a do not have complex multiplication.

Modular form 32368.2.a.a

sage: E.q_eigenform(10)
 
\( q - 2q^{5} - q^{7} - 3q^{9} - 4q^{11} + 2q^{13} - 8q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.