Properties

Label 416955.ce
Number of curves $6$
Conductor $416955$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("416955.ce1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 416955.ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
416955.ce1 416955ce5 [1, 0, 1, -4783258, 4025964581] [2] 14155776 \(\Gamma_0(N)\)-optimal*
416955.ce2 416955ce3 [1, 0, 1, -315883, 55361681] [2, 2] 7077888 \(\Gamma_0(N)\)-optimal*
416955.ce3 416955ce2 [1, 0, 1, -97478, -10946077] [2, 2] 3538944 \(\Gamma_0(N)\)-optimal*
416955.ce4 416955ce1 [1, 0, 1, -95673, -11398049] [2] 1769472 \(\Gamma_0(N)\)-optimal*
416955.ce5 416955ce4 [1, 0, 1, 92047, -48320407] [2] 7077888  
416955.ce6 416955ce6 [1, 0, 1, 657012, 329328913] [2] 14155776  
*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 416955.ce4.

Rank

sage: E.rank()
 

The elliptic curves in class 416955.ce have rank \(1\).

Modular form 416955.2.a.ce

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - q^{7} - 3q^{8} + q^{9} + q^{10} + q^{11} - q^{12} + 2q^{13} - q^{14} + q^{15} - q^{16} - 6q^{17} + q^{18} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.