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Properties

Label 404928.bc
Number of curves $6$
Conductor $404928$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 404928.bc have rank \(0\).

Complex multiplication

The elliptic curves in class 404928.bc do not have complex multiplication.

Modular form 404928.2.a.bc

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{5} + 4 q^{11} + 2 q^{13} - 2 q^{17} - q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 404928.bc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
404928.bc1 404928bc5 \([0, 0, 0, -359576076, 2624425788656]\) \(53809458751271244978434/234099\) \(22368507789312\) \([2]\) \(27262976\) \(3.1298\) \(\Gamma_0(N)\)-optimal*
404928.bc2 404928bc3 \([0, 0, 0, -22473516, 41006609840]\) \(26274189238602645028/54802341801\) \(2618222652485074944\) \([2, 2]\) \(13631488\) \(2.7833\) \(\Gamma_0(N)\)-optimal*
404928.bc3 404928bc6 \([0, 0, 0, -22227276, 41949118064]\) \(-12709983426958940834/600633986620491\) \(-57391471164960006340608\) \([2]\) \(27262976\) \(3.1298\)  
404928.bc4 404928bc2 \([0, 0, 0, -1419996, 625958480]\) \(26511701882112592/1170544394889\) \(13980907337712943104\) \([2, 2]\) \(6815744\) \(2.4367\) \(\Gamma_0(N)\)-optimal*
404928.bc5 404928bc1 \([0, 0, 0, -239016, -32083576]\) \(2022912739489792/574975052397\) \(429216576714150912\) \([2]\) \(3407872\) \(2.0901\) \(\Gamma_0(N)\)-optimal*
404928.bc6 404928bc4 \([0, 0, 0, 737844, 2359998704]\) \(929843593713212/50899738433877\) \(-2431772873083868479488\) \([2]\) \(13631488\) \(2.7833\)  
*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 404928.bc1.