Properties

Label 4032z
Number of curves 6
Conductor 4032
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("4032.r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4032z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4032.r5 4032z1 [0, 0, 0, -300, 4048] [2] 1536 \(\Gamma_0(N)\)-optimal
4032.r4 4032z2 [0, 0, 0, -6060, 181456] [2] 3072  
4032.r6 4032z3 [0, 0, 0, 2580, -84656] [2] 4608  
4032.r3 4032z4 [0, 0, 0, -20460, -923312] [2] 9216  
4032.r2 4032z5 [0, 0, 0, -98220, -11882288] [2] 13824  
4032.r1 4032z6 [0, 0, 0, -1572780, -759189296] [2] 27648  

Rank

sage: E.rank()
 

The elliptic curves in class 4032z have rank \(1\).

Modular form 4032.2.a.r

sage: E.q_eigenform(10)
 
\( q - q^{7} + 4q^{13} - 6q^{17} + 2q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.