Properties

Label 450840s
Number of curves $2$
Conductor $450840$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("s1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 450840s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.s2 450840s1 \([0, -1, 0, -11243600295, -458884411110600]\) \(-82847542748407455193088/141410419921875\) \(-268312982609347534218750000\) \([2]\) \(513884160\) \(4.3321\) \(\Gamma_0(N)\)-optimal*
450840.s1 450840s2 \([0, -1, 0, -179897678420, -29368756943723100]\) \(21208997008348807455199568/61790625\) \(1875870465580144800000\) \([2]\) \(1027768320\) \(4.6786\) \(\Gamma_0(N)\)-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 450840s1.

Rank

sage: E.rank()
 

The elliptic curves in class 450840s have rank \(0\).

Complex multiplication

The elliptic curves in class 450840s do not have complex multiplication.

Modular form 450840.2.a.s

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} - 2q^{7} + q^{9} - 4q^{11} + q^{13} - q^{15} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.