Properties

Label 287490ca
Number of curves $8$
Conductor $287490$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("287490.ca1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 287490ca

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
287490.ca7 287490ca1 [1, 0, 1, -56158, -1807024] [2] 2488320 \(\Gamma_0(N)\)-optimal
287490.ca5 287490ca2 [1, 0, 1, -494238, 132420688] [2, 2] 4976640  
287490.ca4 287490ca3 [1, 0, 1, -3670318, -2706775792] [2] 7464960  
287490.ca2 287490ca4 [1, 0, 1, -7886838, 8524500208] [2] 9953280  
287490.ca6 287490ca5 [1, 0, 1, -110918, 332667056] [2] 9953280  
287490.ca3 287490ca6 [1, 0, 1, -3697698, -2664347744] [2, 2] 14929920  
287490.ca1 287490ca7 [1, 0, 1, -8831448, 6354624256] [2] 29859840  
287490.ca8 287490ca8 [1, 0, 1, 997972, -8967815152] [2] 29859840  

Rank

sage: E.rank()
 

The elliptic curves in class 287490ca have rank \(0\).

Modular form 287490.2.a.ca

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.