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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("w1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 459186.2.a.w

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 3 q^{10} + 3 q^{11} - q^{12} + q^{13} - q^{14} - 3 q^{15} + q^{16} - 6 q^{17} - q^{18} - 5 q^{19} + O(q^{20})\) 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 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 459186w

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
459186.w2 459186w1 \([1, 1, 0, -39324336, -94016625408]\) \(11306285207521130737/125669260394496\) \(74751006815467880841216\) \([]\) \(77898240\) \(3.2033\) \(\Gamma_0(N)\)-optimal
459186.w1 459186w2 \([1, 1, 0, -3176725296, -68916933794688]\) \(5960390431020838621741297/8391766656\) \(4991618511378984576\) \([]\) \(233694720\) \(3.7526\)