Properties

Label 400752.ey
Number of curves $6$
Conductor $400752$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ey1")
 
E.isogeny_class()
 

Elliptic curves in class 400752.ey

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400752.ey1 400752ey5 \([0, 0, 0, -1622244459, 25149106658522]\) \(89254274298475942657/17457\) \(92344960347475968\) \([4]\) \(62914560\) \(3.5565\) \(\Gamma_0(N)\)-optimal*
400752.ey2 400752ey3 \([0, 0, 0, -101390619, 392952021770]\) \(21790813729717297/304746849\) \(1612065972785887973376\) \([2, 2]\) \(31457280\) \(3.2099\) \(\Gamma_0(N)\)-optimal*
400752.ey3 400752ey6 \([0, 0, 0, -98515659, 416284622138]\) \(-19989223566735457/2584262514273\) \(-13670368299708353829015552\) \([2]\) \(62914560\) \(3.5565\)  
400752.ey4 400752ey4 \([0, 0, 0, -24550779, -40606896022]\) \(309368403125137/44372288367\) \(234722873906019600887808\) \([2]\) \(31457280\) \(3.2099\)  
400752.ey5 400752ey2 \([0, 0, 0, -6516939, 5772533690]\) \(5786435182177/627352209\) \(3318600840007243862016\) \([2, 2]\) \(15728640\) \(2.8634\) \(\Gamma_0(N)\)-optimal*
400752.ey6 400752ey1 \([0, 0, 0, 539781, 447532778]\) \(3288008303/18259263\) \(-96588813525183885312\) \([2]\) \(7864320\) \(2.5168\) \(\Gamma_0(N)\)-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 400752.ey1.

Rank

sage: E.rank()
 

The elliptic curves in class 400752.ey have rank \(0\).

Complex multiplication

The elliptic curves in class 400752.ey do not have complex multiplication.

Modular form 400752.2.a.ey

sage: E.q_eigenform(10)
 
\(q + 2 q^{5} + 2 q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.