Properties

Label 287490bc
Number of curves $8$
Conductor $287490$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("287490.bc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 287490bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
287490.bc7 287490bc1 [1, 0, 1, 287461, 44725286] [2] 6488064 \(\Gamma_0(N)\)-optimal
287490.bc6 287490bc2 [1, 0, 1, -1464859, 400095782] [2, 2] 12976128  
287490.bc4 287490bc3 [1, 0, 1, -20630859, 36056522182] [2] 25952256  
287490.bc5 287490bc4 [1, 0, 1, -10335979, -12505609594] [2, 2] 25952256  
287490.bc8 287490bc5 [1, 0, 1, 1738601, -39972864178] [2] 51904512  
287490.bc2 287490bc6 [1, 0, 1, -164348479, -810968014594] [2, 2] 51904512  
287490.bc3 287490bc7 [1, 0, 1, -163321729, -821600626894] [2] 103809024  
287490.bc1 287490bc8 [1, 0, 1, -2629575229, -51901313272294] [2] 103809024  

Rank

sage: E.rank()
 

The elliptic curves in class 287490bc have rank \(2\).

Modular form 287490.2.a.bc

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} + 2q^{13} + q^{14} - q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.