Properties

Label 137904bu
Number of curves 6
Conductor 137904
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("137904.bh1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 137904bu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
137904.bh5 137904bu1 [0, -1, 0, -91992, -9929232] [2] 884736 \(\Gamma_0(N)\)-optimal
137904.bh4 137904bu2 [0, -1, 0, -308312, 54447600] [2, 2] 1769472  
137904.bh2 137904bu3 [0, -1, 0, -4688792, 3909270000] [2, 2] 3538944  
137904.bh6 137904bu4 [0, -1, 0, 611048, 316281328] [2] 3538944  
137904.bh1 137904bu5 [0, -1, 0, -75019832, 250124174832] [2] 7077888  
137904.bh3 137904bu6 [0, -1, 0, -4445432, 4332911088] [2] 7077888  

Rank

sage: E.rank()
 

The elliptic curves in class 137904bu have rank \(1\).

Modular form 137904.2.a.bh

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.