Properties

Label 478800gk
Number of curves $6$
Conductor $478800$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("gk1")
 
E.isogeny_class()
 

Elliptic curves in class 478800gk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
478800.gk4 478800gk1 \([0, 0, 0, -36378075, 84451482250]\) \(114113060120923921/124104960\) \(5790241013760000000\) \([2]\) \(35389440\) \(2.8876\) \(\Gamma_0(N)\)-optimal
478800.gk3 478800gk2 \([0, 0, 0, -36666075, 83046330250]\) \(116844823575501841/3760263939600\) \(175438874365977600000000\) \([2, 2]\) \(70778880\) \(3.2342\)  
478800.gk5 478800gk3 \([0, 0, 0, 11213925, 284477490250]\) \(3342636501165359/751262567039460\) \(-35050906327793045760000000\) \([2]\) \(141557760\) \(3.5808\)  
478800.gk2 478800gk4 \([0, 0, 0, -89154075, -208314557750]\) \(1679731262160129361/570261564022500\) \(26606123531033760000000000\) \([2, 2]\) \(141557760\) \(3.5808\)  
478800.gk6 478800gk5 \([0, 0, 0, 261737925, -1443805289750]\) \(42502666283088696719/43898058864843750\) \(-2048107834398150000000000000\) \([2]\) \(283115520\) \(3.9274\)  
478800.gk1 478800gk6 \([0, 0, 0, -1279854075, -17619920657750]\) \(4969327007303723277361/1123462695162150\) \(52416275505485270400000000\) \([2]\) \(283115520\) \(3.9274\)  

Rank

sage: E.rank()
 

The elliptic curves in class 478800gk have rank \(0\).

Complex multiplication

The elliptic curves in class 478800gk do not have complex multiplication.

Modular form 478800.2.a.gk

sage: E.q_eigenform(10)
 
\(q - q^{7} + 4 q^{11} - 6 q^{13} + 2 q^{17} + q^{19} + O(q^{20})\) 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 Cremona 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

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

The vertices are labelled with Cremona labels.