Properties

Label 42432s
Number of curves $6$
Conductor $42432$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("42432.cf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 42432s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
42432.cf4 42432s1 [0, 1, 0, -34497, 2454687] [2] 65536 \(\Gamma_0(N)\)-optimal
42432.cf3 42432s2 [0, 1, 0, -34817, 2406495] [2, 2] 131072  
42432.cf5 42432s3 [0, 1, 0, 14143, 8683167] [2] 262144  
42432.cf2 42432s4 [0, 1, 0, -88897, -6949345] [2, 2] 262144  
42432.cf6 42432s5 [0, 1, 0, 248063, -46778017] [2] 524288  
42432.cf1 42432s6 [0, 1, 0, -1291137, -565029153] [2] 524288  

Rank

sage: E.rank()
 

The elliptic curves in class 42432s have rank \(1\).

Modular form 42432.2.a.cf

sage: E.q_eigenform(10)
 
\( q + q^{3} + 2q^{5} + q^{9} - 4q^{11} - q^{13} + 2q^{15} + q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.