Properties

Label 33135.d
Number of curves 8
Conductor 33135
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("33135.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 33135.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
33135.d1 33135a8 [1, 1, 1, -4771486, 4009713254] [2] 412160  
33135.d2 33135a6 [1, 1, 1, -298261, 62539514] [2, 2] 206080  
33135.d3 33135a7 [1, 1, 1, -243036, 86485074] [2] 412160  
33135.d4 33135a4 [1, 1, 1, -176766, -28678932] [2] 103040  
33135.d5 33135a3 [1, 1, 1, -22136, 577064] [2, 2] 103040  
33135.d6 33135a2 [1, 1, 1, -11091, -447912] [2, 2] 51520  
33135.d7 33135a1 [1, 1, 1, -46, -19366] [2] 25760 \(\Gamma_0(N)\)-optimal
33135.d8 33135a5 [1, 1, 1, 77269, 4433978] [2] 206080  

Rank

sage: E.rank()
 

The elliptic curves in class 33135.d have rank \(0\).

Modular form 33135.2.a.d

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 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.