# Properties

 Label 450840.x Number of curves $4$ Conductor $450840$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.x1 450840x4 $$[0, -1, 0, -18061440, -29538245700]$$ $$26362547147244676/244298925$$ $$6038304930624844800$$ $$$$ $$21233664$$ $$2.7674$$
450840.x2 450840x2 $$[0, -1, 0, -1154940, -438777900]$$ $$27572037674704/2472575625$$ $$15278582977575840000$$ $$[2, 2]$$ $$10616832$$ $$2.4208$$
450840.x3 450840x1 $$[0, -1, 0, -251815, 40962100]$$ $$4572531595264/776953125$$ $$300060154631250000$$ $$$$ $$5308416$$ $$2.0742$$ $$\Gamma_0(N)$$-optimal*
450840.x4 450840x3 $$[0, -1, 0, 1301560, -2057120100]$$ $$9865576607324/79640206425$$ $$-1968456681223865164800$$ $$$$ $$21233664$$ $$2.7674$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 450840.x1.

## Rank

sage: E.rank()

The elliptic curves in class 450840.x have rank $$1$$.

## Complex multiplication

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

## Modular form 450840.2.a.x

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} - 4q^{11} + q^{13} - q^{15} - 8q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 