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Properties

Label 459186.a
Number of curves $2$
Conductor $459186$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("a1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 459186.a have rank \(0\).

Complex multiplication

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

Modular form 459186.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 459186.a

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
459186.a1 459186a2 \([1, 1, 0, -3729662177, 87668718821205]\) \(235254645262548068204076769349/9342363050119008\) \(227850892429352486112\) \([2]\) \(342558720\) \(3.8441\) \(\Gamma_0(N)\)-optimal*
459186.a2 459186a1 \([1, 1, 0, -233092737, 1369888472565]\) \(-57426975723512548190282309/11445627379633155072\) \(-279147406161873019051008\) \([2]\) \(171279360\) \(3.4975\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 459186.a1.