Properties

Label 487872.cy
Number of curves $2$
Conductor $487872$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cy1")
 
E.isogeny_class()
 

Elliptic curves in class 487872.cy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
487872.cy1 487872cy2 \([0, 0, 0, -1469424, 685579466]\) \(35084566528/1029\) \(10291160662542144\) \([]\) \(7299072\) \(2.1723\) \(\Gamma_0(N)\)-optimal*
487872.cy2 487872cy1 \([0, 0, 0, -31944, -673486]\) \(360448/189\) \(1890213182915904\) \([]\) \(2433024\) \(1.6230\) \(\Gamma_0(N)\)-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.cy1.

Rank

sage: E.rank()
 

The elliptic curves in class 487872.cy have rank \(0\).

Complex multiplication

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

Modular form 487872.2.a.cy

sage: E.q_eigenform(10)
 
\(q - 3 q^{5} + q^{7} + 4 q^{13} + 3 q^{17} - 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.