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Properties

Label 411840.jo
Number of curves $4$
Conductor $411840$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("jo1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 411840.jo have rank \(1\).

Complex multiplication

The elliptic curves in class 411840.jo do not have complex multiplication.

Modular form 411840.2.a.jo

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - q^{11} - q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 411840.jo

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
411840.jo1 411840jo4 \([0, 0, 0, -8504770572, -301885335584784]\) \(355995140004443961140387841/2768480\) \(529064766996480\) \([2]\) \(165150720\) \(3.8472\)  
411840.jo2 411840jo3 \([0, 0, 0, -532746252, -4694626426896]\) \(87501897507774086005761/815991377947460000\) \(155938380716100377640960000\) \([2]\) \(165150720\) \(3.8472\) \(\Gamma_0(N)\)-optimal*
411840.jo3 411840jo2 \([0, 0, 0, -531548172, -4716958158864]\) \(86912881496074271306241/7664481510400\) \(1464705226134414950400\) \([2, 2]\) \(82575360\) \(3.5007\) \(\Gamma_0(N)\)-optimal*
411840.jo4 411840jo1 \([0, 0, 0, -33146892, -74051194896]\) \(-21075830718885163521/199306463150080\) \(-38088058244014622638080\) \([2]\) \(41287680\) \(3.1541\) \(\Gamma_0(N)\)-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 411840.jo1.