Properties

Label 42135.k
Number of curves $8$
Conductor $42135$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 42135.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42135.k1 42135f8 \([1, 0, 1, -6067499, -5753085163]\) \(1114544804970241/405\) \(8976566257245\) \([2]\) \(585728\) \(2.2760\)  
42135.k2 42135f6 \([1, 0, 1, -379274, -89888353]\) \(272223782641/164025\) \(3635509334184225\) \([2, 2]\) \(292864\) \(1.9294\)  
42135.k3 42135f7 \([1, 0, 1, -309049, -124186243]\) \(-147281603041/215233605\) \(-4770515348316540045\) \([2]\) \(585728\) \(2.2760\)  
42135.k4 42135f4 \([1, 0, 1, -224779, 40999811]\) \(56667352321/15\) \(332465416935\) \([2]\) \(146432\) \(1.5829\)  
42135.k5 42135f3 \([1, 0, 1, -28149, -843053]\) \(111284641/50625\) \(1122070782155625\) \([2, 2]\) \(146432\) \(1.5829\)  
42135.k6 42135f2 \([1, 0, 1, -14104, 634481]\) \(13997521/225\) \(4986981254025\) \([2, 2]\) \(73216\) \(1.2363\)  
42135.k7 42135f1 \([1, 0, 1, -59, 27737]\) \(-1/15\) \(-332465416935\) \([2]\) \(36608\) \(0.88972\) \(\Gamma_0(N)\)-optimal
42135.k8 42135f5 \([1, 0, 1, 98256, -6303749]\) \(4733169839/3515625\) \(-77921582094140625\) \([2]\) \(292864\) \(1.9294\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 42135.k do not have complex multiplication.

Modular form 42135.2.a.k

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.