Properties

Label 163254du
Number of curves $6$
Conductor $163254$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("163254.u1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 163254du

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
163254.u5 163254du1 [1, 1, 0, 21291, 2457837] [2] 1179648 \(\Gamma_0(N)\)-optimal
163254.u4 163254du2 [1, 1, 0, -195029, 28286445] [2, 2] 2359296  
163254.u2 163254du3 [1, 1, 0, -2993669, 1992371997] [2] 4718592  
163254.u3 163254du4 [1, 1, 0, -857509, -278441795] [2, 2] 4718592  
163254.u6 163254du5 [1, 1, 0, 1058951, -1344376847] [2] 9437184  
163254.u1 163254du6 [1, 1, 0, -13373649, -18829864503] [2] 9437184  

Rank

sage: E.rank()
 

The elliptic curves in class 163254du have rank \(1\).

Modular form 163254.2.a.u

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - 2q^{10} + 4q^{11} - q^{12} + q^{14} - 2q^{15} + q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.