# Properties

 Label 450840.z Number of curves $2$ Conductor $450840$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.z1 450840z1 $$[0, -1, 0, -81213720, 281308387932]$$ $$2396726313900986596/4154072495625$$ $$102675672570010250880000$$ $$$$ $$53084160$$ $$3.3094$$ $$\Gamma_0(N)$$-optimal
450840.z2 450840z2 $$[0, -1, 0, -55816400, 460420447500]$$ $$-389032340685029858/1627263833203125$$ $$-80441739376936797600000000$$ $$$$ $$106168320$$ $$3.6559$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 450840.2.a.z

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} - 2 q^{11} - q^{13} - q^{15} - 8 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 