# Properties

 Label 450840.a Number of curves $4$ Conductor $450840$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.a1 450840a4 $$[0, -1, 0, -162302496, -795804498180]$$ $$19129597231400697604/26325$$ $$650671620019200$$ $$[2]$$ $$31457280$$ $$3.0087$$
450840.a2 450840a2 $$[0, -1, 0, -10143996, -12431676780]$$ $$18681746265374416/693005625$$ $$4282232599251360000$$ $$[2, 2]$$ $$15728640$$ $$2.6621$$
450840.a3 450840a3 $$[0, -1, 0, -9675816, -13631528484]$$ $$-4053153720264484/903687890625$$ $$-22336336705971600000000$$ $$[2]$$ $$31457280$$ $$3.0087$$
450840.a4 450840a1 $$[0, -1, 0, -663351, -175098924]$$ $$83587439220736/13990184325$$ $$5403024631478494800$$ $$[2]$$ $$7864320$$ $$2.3156$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 450840.a1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 450840.2.a.a

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.