Properties

Label 458640.ld
Number of curves 8
Conductor 458640
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("458640.ld1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 458640.ld

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
458640.ld1 458640ld7 [0, 0, 0, -283583850627, 58125989146260994] [2] 1019215872 \(\Gamma_0(N)\)-optimal*
458640.ld2 458640ld6 [0, 0, 0, -17724001827, 908217379192354] [2, 2] 509607936 \(\Gamma_0(N)\)-optimal*
458640.ld3 458640ld8 [0, 0, 0, -17506183107, 931627794634306] [2] 1019215872  
458640.ld4 458640ld4 [0, 0, 0, -3502633827, 79657407097954] [2] 339738624 \(\Gamma_0(N)\)-optimal*
458640.ld5 458640ld3 [0, 0, 0, -1121374947, 13823907590626] [2] 254803968 \(\Gamma_0(N)\)-optimal*
458640.ld6 458640ld2 [0, 0, 0, -292153827, 339930313954] [2, 2] 169869312 \(\Gamma_0(N)\)-optimal*
458640.ld7 458640ld1 [0, 0, 0, -181515747, -936235683614] [2] 84934656 \(\Gamma_0(N)\)-optimal*
458640.ld8 458640ld5 [0, 0, 0, 1148116893, 2697077374306] [2] 339738624  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 458640.ld7.

Rank

sage: E.rank()
 

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

Modular form 458640.2.a.ld

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.