Properties

Label 159120z
Number of curves 8
Conductor 159120
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("159120.ek1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 159120z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
159120.ek7 159120z1 [0, 0, 0, -301845027, 1943629784354] [2] 53084160 \(\Gamma_0(N)\)-optimal
159120.ek6 159120z2 [0, 0, 0, -800246307, -6128776387294] [2, 2] 106168320  
159120.ek5 159120z3 [0, 0, 0, -3714161187, -86556842855134] [2] 159252480  
159120.ek8 159120z4 [0, 0, 0, 2142837213, -40735318265566] [2] 212336640  
159120.ek4 159120z5 [0, 0, 0, -11717750307, -488156229494494] [2] 212336640  
159120.ek2 159120z6 [0, 0, 0, -59319000867, -5560819946447326] [2, 2] 318504960  
159120.ek3 159120z7 [0, 0, 0, -59211435747, -5581991709347614] [2] 637009920  
159120.ek1 159120z8 [0, 0, 0, -949104000867, -355892486813447326] [2] 637009920  

Rank

sage: E.rank()
 

The elliptic curves in class 159120z have rank \(1\).

Modular form 159120.2.a.ek

sage: E.q_eigenform(10)
 
\( q + q^{5} + 4q^{7} + q^{13} - q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.