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Properties

Label 462722.b
Number of curves $2$
Conductor $462722$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("b1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 462722.b have rank \(0\).

Complex multiplication

The elliptic curves in class 462722.b do not have complex multiplication.

Modular form 462722.2.a.b

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 462722.b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462722.b1 462722b2 \([1, 1, 0, -440846, 114360628]\) \(-1680914269/32768\) \(-184709969365336064\) \([]\) \(6199200\) \(2.1066\) \(\Gamma_0(N)\)-optimal*
462722.b2 462722b1 \([1, 1, 0, 4079, -305443]\) \(1331/8\) \(-45095207364584\) \([]\) \(1239840\) \(1.3019\) \(\Gamma_0(N)\)-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 462722.b1.