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Properties

Label 487872.qr
Number of curves $2$
Conductor $487872$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("qr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 487872.qr have rank \(1\).

Complex multiplication

The elliptic curves in class 487872.qr do not have complex multiplication.

Modular form 487872.2.a.qr

Copy content sage:E.q_eigenform(10)
 
\(q + 4 q^{5} - q^{7} + 6 q^{13} - 2 q^{19} + O(q^{20})\) 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 487872.qr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
487872.qr1 487872qr2 \([0, 0, 0, -419628, -81670160]\) \(193100552/43659\) \(1847597467519254528\) \([2]\) \(9830400\) \(2.2178\) \(\Gamma_0(N)\)-optimal*
487872.qr2 487872qr1 \([0, 0, 0, 59532, -7879520]\) \(4410944/7623\) \(-40324547902205952\) \([2]\) \(4915200\) \(1.8712\) \(\Gamma_0(N)\)-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 487872.qr1.