Properties

Label 422370dw
Number of curves $6$
Conductor $422370$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("422370.dw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 422370dw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
422370.dw6 422370dw1 [1, -1, 1, 48667, 1642781] [2] 3538944 \(\Gamma_0(N)\)-optimal*
422370.dw5 422370dw2 [1, -1, 1, -211253, 13807037] [2, 2] 7077888 \(\Gamma_0(N)\)-optimal*
422370.dw2 422370dw3 [1, -1, 1, -2745473, 1750254581] [2] 14155776 \(\Gamma_0(N)\)-optimal*
422370.dw3 422370dw4 [1, -1, 1, -1835753, -947247163] [2, 2] 14155776  
422370.dw4 422370dw5 [1, -1, 1, -373703, -2415730183] [2] 28311552  
422370.dw1 422370dw6 [1, -1, 1, -29289803, -61005726943] [2] 28311552  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 422370dw1.

Rank

sage: E.rank()
 

The elliptic curves in class 422370dw have rank \(0\).

Modular form 422370.2.a.dw

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} - 4q^{11} - q^{13} + q^{16} + 6q^{17} + O(q^{20}) \)

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.