Properties

Label 406560ch
Number of curves $4$
Conductor $406560$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("ch1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 406560ch

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
406560.ch3 406560ch1 \([0, -1, 0, -771828790, -8253078202400]\) \(448487713888272974160064/91549016015625\) \(10379818647146025000000\) \([2, 2]\) \(137625600\) \(3.6124\) \(\Gamma_0(N)\)-optimal*
406560.ch2 406560ch2 \([0, -1, 0, -774475665, -8193619331775]\) \(7079962908642659949376/100085966990454375\) \(726255189063992701048320000\) \([2]\) \(275251200\) \(3.9589\) \(\Gamma_0(N)\)-optimal*
406560.ch4 406560ch3 \([0, -1, 0, -769182520, -8312482729868]\) \(-55486311952875723077768/801237030029296875\) \(-726753420367734375000000000\) \([2]\) \(275251200\) \(3.9589\)  
406560.ch1 406560ch4 \([0, -1, 0, -12349260040, -528209408557400]\) \(229625675762164624948320008/9568125\) \(8678664751680000\) \([2]\) \(275251200\) \(3.9589\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 406560ch1.

Rank

sage: E.rank()
 

The elliptic curves in class 406560ch have rank \(1\).

Complex multiplication

The elliptic curves in class 406560ch do not have complex multiplication.

Modular form 406560.2.a.ch

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} - q^{7} + q^{9} + 6q^{13} - q^{15} - 6q^{17} + 4q^{19} + O(q^{20})\)  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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.