Properties

Label 481650.hr
Number of curves $4$
Conductor $481650$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("hr1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 481650.hr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
481650.hr1 481650hr4 \([1, 0, 0, -2623813, -1633705633]\) \(26487576322129/44531250\) \(3358497473144531250\) \([2]\) \(14155776\) \(2.4501\)  
481650.hr2 481650hr2 \([1, 0, 0, -215563, -8136883]\) \(14688124849/8122500\) \(612589939101562500\) \([2, 2]\) \(7077888\) \(2.1035\)  
481650.hr3 481650hr1 \([1, 0, 0, -131063, 18142617]\) \(3301293169/22800\) \(1719550706250000\) \([2]\) \(3538944\) \(1.7570\) \(\Gamma_0(N)\)-optimal*
481650.hr4 481650hr3 \([1, 0, 0, 840687, -64118133]\) \(871257511151/527800050\) \(-39806094242819531250\) \([2]\) \(14155776\) \(2.4501\)  
*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.hr1.

Rank

sage: E.rank()
 

The elliptic curves in class 481650.hr have rank \(0\).

Complex multiplication

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

Modular form 481650.2.a.hr

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