Properties

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

Related objects

Downloads

Learn more

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

Elliptic curves in class 416955.bs

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
416955.bs1 416955bs5 \([1, 1, 0, -1100761207, 14056384132276]\) \(3135316978843283198764801/571725\) \(26897306314725\) \([2]\) \(53084160\) \(3.3714\) \(\Gamma_0(N)\)-optimal*
416955.bs2 416955bs3 \([1, 1, 0, -68797582, 219609455551]\) \(765458482133960722801/326869475625\) \(15377862452786150625\) \([2, 2]\) \(26542080\) \(3.0248\) \(\Gamma_0(N)\)-optimal*
416955.bs3 416955bs6 \([1, 1, 0, -68456437, 221895604654]\) \(-754127868744065783521/15825714261328125\) \(-744534669878445870703125\) \([2]\) \(53084160\) \(3.3714\)  
416955.bs4 416955bs4 \([1, 1, 0, -9185652, -5654617371]\) \(1821931919215868881/761147600816295\) \(35808859451438917430895\) \([2]\) \(26542080\) \(3.0248\)  
416955.bs5 416955bs2 \([1, 1, 0, -4321177, 3394279024]\) \(189674274234120481/3859869269025\) \(181590950306107136025\) \([2, 2]\) \(13271040\) \(2.6782\) \(\Gamma_0(N)\)-optimal*
416955.bs6 416955bs1 \([1, 1, 0, 12628, 158660211]\) \(4733169839/231139696095\) \(-10874170636861534695\) \([2]\) \(6635520\) \(2.3316\) \(\Gamma_0(N)\)-optimal*
*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 416955.bs1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 416955.2.a.bs

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.