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Properties

Label 424830fk
Number of curves $8$
Conductor $424830$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("fk1")
 
E.isogeny_class()
 

Elliptic curves in class 424830fk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424830.fk6 424830fk1 \([1, 1, 1, -1133175, -570931635]\) \(-56667352321/16711680\) \(-47457174689982382080\) \([2]\) \(14155776\) \(2.4900\) \(\Gamma_0(N)\)-optimal*
424830.fk5 424830fk2 \([1, 1, 1, -19259255, -32538086323]\) \(278202094583041/16646400\) \(47271795101349638400\) \([2, 2]\) \(28311552\) \(2.8366\) \(\Gamma_0(N)\)-optimal*
424830.fk4 424830fk3 \([1, 1, 1, -20392135, -28496423635]\) \(330240275458561/67652010000\) \(192115529779078764810000\) \([2, 2]\) \(56623104\) \(3.1832\) \(\Gamma_0(N)\)-optimal*
424830.fk2 424830fk4 \([1, 1, 1, -308143655, -2082115127443]\) \(1139466686381936641/4080\) \(11586224289546480\) \([2]\) \(56623104\) \(3.1832\)  
424830.fk3 424830fk5 \([1, 1, 1, -102242715, 372604158597]\) \(41623544884956481/2962701562500\) \(8413363923067355076562500\) \([2, 2]\) \(113246208\) \(3.5297\) \(\Gamma_0(N)\)-optimal*
424830.fk7 424830fk6 \([1, 1, 1, 43332365, -170907936235]\) \(3168685387909439/6278181696900\) \(-17828534625198263906328900\) \([2]\) \(113246208\) \(3.5297\)  
424830.fk1 424830fk7 \([1, 1, 1, -1606848965, 24791159911097]\) \(161572377633716256481/914742821250\) \(2597650856435054901521250\) \([2]\) \(226492416\) \(3.8763\) \(\Gamma_0(N)\)-optimal*
424830.fk8 424830fk8 \([1, 1, 1, 92754255, 1626122680545]\) \(31077313442863199/420227050781250\) \(-1193344329138774719238281250\) \([2]\) \(226492416\) \(3.8763\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 5 curves highlighted, and conditionally curve 424830fk1.

Rank

sage: E.rank()
 

The elliptic curves in class 424830fk have rank \(0\).

Complex multiplication

The elliptic curves in class 424830fk do not have complex multiplication.

Modular form 424830.2.a.fk

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} - 4 q^{11} - q^{12} + 2 q^{13} - q^{15} + q^{16} + q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.