Properties

Label 485520.gt
Number of curves $4$
Conductor $485520$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("gt1")
 
E.isogeny_class()
 

Elliptic curves in class 485520.gt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485520.gt1 485520gt4 \([0, 1, 0, -837040, -295006972]\) \(2624033547076/324135\) \(8011602870082560\) \([2]\) \(7864320\) \(2.0749\)  
485520.gt2 485520gt2 \([0, 1, 0, -56740, -3799012]\) \(3269383504/893025\) \(5518195854393600\) \([2, 2]\) \(3932160\) \(1.7283\)  
485520.gt3 485520gt1 \([0, 1, 0, -20615, 1085088]\) \(2508888064/118125\) \(45620005410000\) \([2]\) \(1966080\) \(1.3817\) \(\Gamma_0(N)\)-optimal*
485520.gt4 485520gt3 \([0, 1, 0, 145560, -24595452]\) \(13799183324/18600435\) \(-459744546040335360\) \([2]\) \(7864320\) \(2.0749\)  
*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 485520.gt1.

Rank

sage: E.rank()
 

The elliptic curves in class 485520.gt have rank \(1\).

Complex multiplication

The elliptic curves in class 485520.gt do not have complex multiplication.

Modular form 485520.2.a.gt

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} - q^{7} + q^{9} - 4 q^{11} - 6 q^{13} + q^{15} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.