Properties

Label 485100.cn
Number of curves $4$
Conductor $485100$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 485100.cn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485100.cn1 485100cn4 \([0, 0, 0, -2756209575, -55695033427250]\) \(6749703004355978704/5671875\) \(1945818870187500000000\) \([2]\) \(119439360\) \(3.8205\)  
485100.cn2 485100cn3 \([0, 0, 0, -172225200, -870636942875]\) \(-26348629355659264/24169921875\) \(-518240111022949218750000\) \([2]\) \(59719680\) \(3.4740\)  
485100.cn3 485100cn2 \([0, 0, 0, -34798575, -72754330250]\) \(13584145739344/1195803675\) \(410237770729178700000000\) \([2]\) \(39813120\) \(3.2712\) \(\Gamma_0(N)\)-optimal*
485100.cn4 485100cn1 \([0, 0, 0, 2410800, -5293733375]\) \(72268906496/606436875\) \(-13002934600027968750000\) \([2]\) \(19906560\) \(2.9247\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 485100.cn1.

Rank

sage: E.rank()
 

The elliptic curves in class 485100.cn have rank \(0\).

Complex multiplication

The elliptic curves in class 485100.cn do not have complex multiplication.

Modular form 485100.2.a.cn

sage: E.q_eigenform(10)
 
\(q - q^{11} + 2q^{13} - 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.