Properties

Label 446400ih
Number of curves $6$
Conductor $446400$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("446400.ih1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 446400ih

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
446400.ih6 446400ih1 [0, 0, 0, 863700, 1750462000] [2] 18874368 \(\Gamma_0(N)\)-optimal*
446400.ih5 446400ih2 [0, 0, 0, -17568300, 26781118000] [2, 2] 37748736 \(\Gamma_0(N)\)-optimal*
446400.ih2 446400ih3 [0, 0, 0, -276768300, 1772233918000] [2, 2] 75497472 \(\Gamma_0(N)\)-optimal*
446400.ih4 446400ih4 [0, 0, 0, -53280300, -116709698000] [2] 75497472  
446400.ih1 446400ih5 [0, 0, 0, -4428288300, 113423212798000] [2] 150994944 \(\Gamma_0(N)\)-optimal*
446400.ih3 446400ih6 [0, 0, 0, -272448300, 1830234238000] [2] 150994944  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 446400ih1.

Rank

sage: E.rank()
 

The elliptic curves in class 446400ih have rank \(0\).

Modular form 446400.2.a.ih

sage: E.q_eigenform(10)
 
\( q - 4q^{11} + 6q^{13} + 2q^{17} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.