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Properties

Label 459186.q
Number of curves $2$
Conductor $459186$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("q1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 459186.q have rank \(1\).

Complex multiplication

The elliptic curves in class 459186.q do not have complex multiplication.

Modular form 459186.2.a.q

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + 3 q^{11} - q^{12} + q^{13} - q^{14} - q^{15} + q^{16} + 2 q^{17} - q^{18} + 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 459186.q

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
459186.q1 459186q2 \([1, 1, 0, -16391998297, -807792164146877]\) \(818901045522640857815176321/11199087354\) \(6661478332075382634\) \([]\) \(336000000\) \(4.0993\)  
459186.q2 459186q1 \([1, 1, 0, -28905187, -40512120467]\) \(4490173250235298081/1407217448486304\) \(837045756077769768295584\) \([]\) \(67200000\) \(3.2945\) \(\Gamma_0(N)\)-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 459186.q1.