Properties

Label 426888gy
Number of curves $4$
Conductor $426888$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("gy1")
 
E.isogeny_class()
 

Elliptic curves in class 426888gy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
426888.gy4 426888gy1 \([0, 0, 0, 1476321, 27330804578]\) \(9148592/8301447\) \(-322898214978979871438592\) \([2]\) \(47185920\) \(3.1900\) \(\Gamma_0(N)\)-optimal*
426888.gy3 426888gy2 \([0, 0, 0, -127657299, 542393161310]\) \(1478729816932/38900169\) \(6052340095639060730766336\) \([2, 2]\) \(94371840\) \(3.5366\) \(\Gamma_0(N)\)-optimal*
426888.gy1 426888gy3 \([0, 0, 0, -2029443339, 35189511595238]\) \(2970658109581346/2139291\) \(665689483022029126391808\) \([2]\) \(188743680\) \(3.8831\) \(\Gamma_0(N)\)-optimal*
426888.gy2 426888gy4 \([0, 0, 0, -292009179, -1140734441770]\) \(8849350367426/3314597517\) \(1031413074480204609209296896\) \([2]\) \(188743680\) \(3.8831\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 426888gy1.

Rank

sage: E.rank()
 

The elliptic curves in class 426888gy have rank \(1\).

Complex multiplication

The elliptic curves in class 426888gy do not have complex multiplication.

Modular form 426888.2.a.gy

sage: E.q_eigenform(10)
 
\(q + 2 q^{5} + 6 q^{13} - 6 q^{17} + O(q^{20})\) Copy content 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 Cremona 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 Cremona labels.