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Properties

Label 417450.fo
Number of curves $6$
Conductor $417450$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("fo1")
 
E.isogeny_class()
 

Elliptic curves in class 417450.fo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
417450.fo1 417450fo4 \([1, 1, 1, -333960063, -2349176461719]\) \(148809678420065817601/20700\) \(572989260937500\) \([2]\) \(47185920\) \(3.1567\)  
417450.fo2 417450fo5 \([1, 1, 1, -78135813, 227684489781]\) \(1905890658841300321/293666194803750\) \(8128868402073846152343750\) \([2]\) \(94371840\) \(3.5033\) \(\Gamma_0(N)\)-optimal*
417450.fo3 417450fo3 \([1, 1, 1, -21417063, -34696447719]\) \(39248884582600321/3935264062500\) \(108930630278540039062500\) \([2, 2]\) \(47185920\) \(3.1567\) \(\Gamma_0(N)\)-optimal*
417450.fo4 417450fo2 \([1, 1, 1, -20872563, -36712186719]\) \(36330796409313601/428490000\) \(11860877701406250000\) \([2, 2]\) \(23592960\) \(2.8101\) \(\Gamma_0(N)\)-optimal*
417450.fo5 417450fo1 \([1, 1, 1, -1270563, -605302719]\) \(-8194759433281/965779200\) \(-26733386958300000000\) \([2]\) \(11796480\) \(2.4635\) \(\Gamma_0(N)\)-optimal*
417450.fo6 417450fo6 \([1, 1, 1, 26589687, -168059199219]\) \(75108181893694559/484313964843750\) \(-13406120810508728027343750\) \([2]\) \(94371840\) \(3.5033\)  
*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 417450.fo1.

Rank

sage: E.rank()
 

The elliptic curves in class 417450.fo have rank \(0\).

Complex multiplication

The elliptic curves in class 417450.fo do not have complex multiplication.

Modular form 417450.2.a.fo

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} - q^{12} - 2 q^{13} + q^{16} - 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.