Properties

Label 83490bb
Number of curves $6$
Conductor $83490$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("83490.bc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 83490bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
83490.bc5 83490bb1 [1, 0, 1, -50823, -4842422] [2] 491520 \(\Gamma_0(N)\)-optimal
83490.bc4 83490bb2 [1, 0, 1, -834903, -293697494] [2, 2] 983040  
83490.bc3 83490bb3 [1, 0, 1, -856683, -277571582] [2, 2] 1966080  
83490.bc1 83490bb4 [1, 0, 1, -13358403, -18793411694] [2] 1966080  
83490.bc6 83490bb5 [1, 0, 1, 1063587, -1344473594] [2] 3932160  
83490.bc2 83490bb6 [1, 0, 1, -3125433, 1821475918] [2] 3932160  

Rank

sage: E.rank()
 

The elliptic curves in class 83490bb have rank \(0\).

Modular form 83490.2.a.bc

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.