Properties

Label 416955bs
Number of curves $6$
Conductor $416955$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("bs1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 416955bs

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
416955.bs6 416955bs1 [1, 1, 0, 12628, 158660211] [2] 6635520 \(\Gamma_0(N)\)-optimal*
416955.bs5 416955bs2 [1, 1, 0, -4321177, 3394279024] [2, 2] 13271040 \(\Gamma_0(N)\)-optimal*
416955.bs2 416955bs3 [1, 1, 0, -68797582, 219609455551] [2, 2] 26542080 \(\Gamma_0(N)\)-optimal*
416955.bs4 416955bs4 [1, 1, 0, -9185652, -5654617371] [2] 26542080  
416955.bs1 416955bs5 [1, 1, 0, -1100761207, 14056384132276] [2] 53084160 \(\Gamma_0(N)\)-optimal*
416955.bs3 416955bs6 [1, 1, 0, -68456437, 221895604654] [2] 53084160  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 416955bs1.

Rank

sage: E.rank()
 

The elliptic curves in class 416955bs have rank \(0\).

Complex multiplication

The elliptic curves in class 416955bs do not have complex multiplication.

Modular form 416955.2.a.bs

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.