Properties

Label 485520.hz
Number of curves 4
Conductor 485520
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("485520.hz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 485520.hz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
485520.hz1 485520hz4 [0, 1, 0, -7796160, -8380796940] [2] 18874368  
485520.hz2 485520hz3 [0, 1, 0, -2478560, 1395356340] [4] 18874368 \(\Gamma_0(N)\)-optimal*
485520.hz3 485520hz2 [0, 1, 0, -513360, -116275500] [2, 2] 9437184 \(\Gamma_0(N)\)-optimal*
485520.hz4 485520hz1 [0, 1, 0, 64640, -10617100] [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 0 curves highlighted, and conditionally curve 485520.hz4.

Rank

sage: E.rank()
 

The elliptic curves in class 485520.hz have rank \(1\).

Modular form 485520.2.a.hz

sage: E.q_eigenform(10)
 
\( q + q^{3} + q^{5} + q^{7} + q^{9} - 6q^{13} + q^{15} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.