Properties

Label 13005.p
Number of curves 8
Conductor 13005
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("13005.p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 13005.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
13005.p1 13005n7 [1, -1, 0, -5618214, 5127014475] [2] 163840  
13005.p2 13005n5 [1, -1, 0, -351189, 80151120] [2, 2] 81920  
13005.p3 13005n8 [1, -1, 0, -286164, 110699865] [2] 163840  
13005.p4 13005n3 [1, -1, 0, -208134, -36495927] [2] 40960  
13005.p5 13005n4 [1, -1, 0, -26064, 755595] [2, 2] 40960  
13005.p6 13005n2 [1, -1, 0, -13059, -563112] [2, 2] 20480  
13005.p7 13005n1 [1, -1, 0, -54, -24705] [2] 10240 \(\Gamma_0(N)\)-optimal
13005.p8 13005n6 [1, -1, 0, 90981, 5601258] [2] 81920  

Rank

sage: E.rank()
 

The elliptic curves in class 13005.p have rank \(1\).

Modular form 13005.2.a.p

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

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.