Properties

Label 455175ce
Number of curves 4
Conductor 455175
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("455175.ce1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 455175ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
455175.ce4 455175ce1 [1, -1, 1, 908995, -561249628] [2] 14155776 \(\Gamma_0(N)\)-optimal*
455175.ce3 455175ce2 [1, -1, 1, -7219130, -6120887128] [2, 2] 28311552 \(\Gamma_0(N)\)-optimal*
455175.ce2 455175ce3 [1, -1, 1, -34854755, 73635526622] [2] 56623104 \(\Gamma_0(N)\)-optimal*
455175.ce1 455175ce4 [1, -1, 1, -109633505, -441791638378] [2] 56623104  
*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 455175ce1.

Rank

sage: E.rank()
 

The elliptic curves in class 455175ce have rank \(1\).

Modular form 455175.2.a.ce

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.