Properties

Label 42135f
Number of curves 8
Conductor 42135
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("42135.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 42135f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
42135.k7 42135f1 [1, 0, 1, -59, 27737] [2] 36608 \(\Gamma_0(N)\)-optimal
42135.k6 42135f2 [1, 0, 1, -14104, 634481] [2, 2] 73216  
42135.k5 42135f3 [1, 0, 1, -28149, -843053] [2, 2] 146432  
42135.k4 42135f4 [1, 0, 1, -224779, 40999811] [2] 146432  
42135.k8 42135f5 [1, 0, 1, 98256, -6303749] [2] 292864  
42135.k2 42135f6 [1, 0, 1, -379274, -89888353] [2, 2] 292864  
42135.k3 42135f7 [1, 0, 1, -309049, -124186243] [2] 585728  
42135.k1 42135f8 [1, 0, 1, -6067499, -5753085163] [2] 585728  

Rank

sage: E.rank()
 

The elliptic curves in class 42135f have rank \(0\).

Modular form 42135.2.a.k

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} - 2q^{13} - q^{15} - q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.