Properties

Label 458640.na
Number of curves $6$
Conductor $458640$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("458640.na1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 458640.na

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
458640.na1 458640na5 [0, 0, 0, -63609987, 195270223106] [2] 37748736 \(\Gamma_0(N)\)-optimal*
458640.na2 458640na4 [0, 0, 0, -5962467, -5599434526] [2] 18874368  
458640.na3 458640na3 [0, 0, 0, -3986787, 3033101666] [2, 2] 18874368 \(\Gamma_0(N)\)-optimal*
458640.na4 458640na6 [0, 0, 0, -811587, 7731762626] [2] 37748736  
458640.na5 458640na2 [0, 0, 0, -458787, -44019934] [2, 2] 9437184 \(\Gamma_0(N)\)-optimal*
458640.na6 458640na1 [0, 0, 0, 105693, -5296606] [2] 4718592 \(\Gamma_0(N)\)-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 458640.na6.

Rank

sage: E.rank()
 

The elliptic curves in class 458640.na have rank \(1\).

Modular form 458640.2.a.na

sage: E.q_eigenform(10)
 
\( q + q^{5} + 4q^{11} - q^{13} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.