Properties

Label 139638bd
Number of curves 6
Conductor 139638
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("139638.o1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 139638bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
139638.o5 139638bd1 [1, 0, 1, -46575, 3584098] [2] 829440 \(\Gamma_0(N)\)-optimal
139638.o4 139638bd2 [1, 0, 1, -156095, -19590334] [2, 2] 1658880  
139638.o6 139638bd3 [1, 0, 1, 309365, -113985622] [2] 3317760  
139638.o2 139638bd4 [1, 0, 1, -2373875, -1407920614] [2, 2] 3317760  
139638.o3 139638bd5 [1, 0, 1, -2250665, -1560553162] [2] 6635520  
139638.o1 139638bd6 [1, 0, 1, -37981565, -90099554866] [2] 6635520  

Rank

sage: E.rank()
 

The elliptic curves in class 139638bd have rank \(0\).

Modular form 139638.2.a.o

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} - q^{8} + q^{9} - 2q^{10} - 4q^{11} + q^{12} + 2q^{13} + 2q^{15} + q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.