Properties

Label 139425n
Number of curves $6$
Conductor $139425$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("139425.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 139425n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
139425.f5 139425n1 [1, 1, 1, -101488, 17808656] [2] 1376256 \(\Gamma_0(N)\)-optimal
139425.f4 139425n2 [1, 1, 1, -1812613, 938393906] [2, 2] 2752512  
139425.f1 139425n3 [1, 1, 1, -29000488, 60099209906] [2] 5505024  
139425.f3 139425n4 [1, 1, 1, -2002738, 729256406] [2, 2] 5505024  
139425.f6 139425n5 [1, 1, 1, 5665637, 4977536156] [2] 11010048  
139425.f2 139425n6 [1, 1, 1, -12713113, -16900020844] [2] 11010048  

Rank

sage: E.rank()
 

The elliptic curves in class 139425n have rank \(0\).

Modular form 139425.2.a.f

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

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.