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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 459186.2.a.d

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 459186d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
459186.d2 459186d1 \([1, 1, 0, -262409, 14572869]\) \(3359498792833/1787595264\) \(1063303351536351744\) \([]\) \(8709120\) \(2.1504\) \(\Gamma_0(N)\)-optimal
459186.d1 459186d2 \([1, 1, 0, -12271889, -16551586419]\) \(343613355239411713/9001882344\) \(5354529551109344424\) \([]\) \(26127360\) \(2.6997\)