# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("h1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.h1 450840h1 $$[0, -1, 0, -4771, 128440]$$ $$152818608128/7605$$ $$597813840$$ $$$$ $$409600$$ $$0.75529$$ $$\Gamma_0(N)$$-optimal
450840.h2 450840h2 $$[0, -1, 0, -4516, 142516]$$ $$-8100185168/2142075$$ $$-2694147705600$$ $$$$ $$819200$$ $$1.1019$$

## Rank

sage: E.rank()

The elliptic curves in class 450840.h have rank $$2$$.

## Complex multiplication

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

## Modular form 450840.2.a.h

sage: E.q_eigenform(10)

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