Properties

Label 13650.d
Number of curves 8
Conductor 13650
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("13650.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 13650.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
13650.d1 13650a7 [1, 1, 0, -1004761375, -12259064860625] [2] 2654208  
13650.d2 13650a6 [1, 1, 0, -62797625, -191567259375] [2, 2] 1327104  
13650.d3 13650a8 [1, 1, 0, -62025875, -196504144125] [2] 2654208  
13650.d4 13650a4 [1, 1, 0, -12410125, -16804746875] [2] 884736  
13650.d5 13650a3 [1, 1, 0, -3973125, -2917087875] [2] 663552  
13650.d6 13650a2 [1, 1, 0, -1035125, -72121875] [2, 2] 442368  
13650.d7 13650a1 [1, 1, 0, -643125, 197182125] [2] 221184 \(\Gamma_0(N)\)-optimal
13650.d8 13650a5 [1, 1, 0, 4067875, -567112875] [2] 884736  

Rank

sage: E.rank()
 

The elliptic curves in class 13650.d have rank \(1\).

Modular form 13650.2.a.d

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} - q^{12} - q^{13} + q^{14} + q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.