# Properties

 Label 450840.s Number of curves $2$ Conductor $450840$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 450840.s

sage: E.isogeny_class().curves

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

## Rank

sage: E.rank()

The elliptic curves in class 450840.s have rank $$0$$.

## Complex multiplication

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

## Modular form 450840.2.a.s

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} - 2q^{7} + q^{9} - 4q^{11} + q^{13} - q^{15} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 