Properties

Label 481650.fd
Number of curves $4$
Conductor $481650$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fd1")
 
E.isogeny_class()
 

Elliptic curves in class 481650.fd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
481650.fd1 481650fd4 \([1, 1, 1, -223179055213, -40581608879320969]\) \(16300610738133468173382620881/2228489100\) \(168070175691904687500\) \([2]\) \(1244160000\) \(4.6960\)  
481650.fd2 481650fd3 \([1, 1, 1, -13948689713, -634092115867969]\) \(-3979640234041473454886161/1471455901872240\) \(-110975571722813201283750000\) \([2]\) \(622080000\) \(4.3494\)  
481650.fd3 481650fd2 \([1, 1, 1, -371567713, -2375289295969]\) \(75224183150104868881/11219310000000000\) \(846147913777968750000000000\) \([2]\) \(248832000\) \(3.8913\) \(\Gamma_0(N)\)-optimal*
481650.fd4 481650fd1 \([1, 1, 1, 39440287, -202701007969]\) \(89962967236397039/287450726400000\) \(-21679214894438400000000000\) \([2]\) \(124416000\) \(3.5447\) \(\Gamma_0(N)\)-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 481650.fd1.

Rank

sage: E.rank()
 

The elliptic curves in class 481650.fd have rank \(1\).

Complex multiplication

The elliptic curves in class 481650.fd do not have complex multiplication.

Modular form 481650.2.a.fd

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - 2 q^{7} + q^{8} + q^{9} - 2 q^{11} - q^{12} - 2 q^{14} + q^{16} + 2 q^{17} + q^{18} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.