Properties

Label 19600cp
Number of curves 6
Conductor 19600
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("19600.dl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 19600cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19600.dl5 19600cp1 [0, -1, 0, -10208, 806912] [2] 55296 \(\Gamma_0(N)\)-optimal
19600.dl4 19600cp2 [0, -1, 0, -206208, 36086912] [2] 110592  
19600.dl6 19600cp3 [0, -1, 0, 87792, -16833088] [2] 165888  
19600.dl3 19600cp4 [0, -1, 0, -696208, -183041088] [2] 331776  
19600.dl2 19600cp5 [0, -1, 0, -3342208, -2357465088] [2] 497664  
19600.dl1 19600cp6 [0, -1, 0, -53518208, -150677721088] [2] 995328  

Rank

sage: E.rank()
 

The elliptic curves in class 19600cp have rank \(1\).

Modular form 19600.2.a.dl

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.