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Properties

Label 485520.bg
Number of curves $6$
Conductor $485520$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bg1")
 
E.isogeny_class()
 

Elliptic curves in class 485520.bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485520.bg1 485520bg5 \([0, -1, 0, -8057690016, 278398954932480]\) \(585196747116290735872321/836876053125000\) \(82739828640777638400000000\) \([2]\) \(509607936\) \(4.2444\) \(\Gamma_0(N)\)-optimal*
485520.bg2 485520bg4 \([0, -1, 0, -1168114976, -15363721416960]\) \(1782900110862842086081/328139630024640\) \(32442339169706883891855360\) \([2]\) \(254803968\) \(3.8978\)  
485520.bg3 485520bg3 \([0, -1, 0, -508177696, 4267103178496]\) \(146796951366228945601/5397929064360000\) \(533679657976196877680640000\) \([2, 2]\) \(254803968\) \(3.8978\) \(\Gamma_0(N)\)-optimal*
485520.bg4 485520bg2 \([0, -1, 0, -80550176, -187407171840]\) \(584614687782041281/184812061593600\) \(18271903288310865749606400\) \([2, 2]\) \(127401984\) \(3.5513\) \(\Gamma_0(N)\)-optimal*
485520.bg5 485520bg1 \([0, -1, 0, 14149344, -19902660864]\) \(3168685387909439/3563732336640\) \(-352337244869342118543360\) \([2]\) \(63700992\) \(3.2047\) \(\Gamma_0(N)\)-optimal*
485520.bg6 485520bg6 \([0, -1, 0, 199294304, 15217071805696]\) \(8854313460877886399/1016927675429790600\) \(-100541079280419533877458534400\) \([2]\) \(509607936\) \(4.2444\)  
*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 485520.bg1.

Rank

sage: E.rank()
 

The elliptic curves in class 485520.bg have rank \(0\).

Complex multiplication

The elliptic curves in class 485520.bg do not have complex multiplication.

Modular form 485520.2.a.bg

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} + 4 q^{11} + 6 q^{13} + q^{15} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.