# Properties

 Label 585.h Number of curves $2$ Conductor $585$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 585.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.h1 585h1 $$[1, -1, 0, -9, 0]$$ $$117649/65$$ $$47385$$ $$$$ $$48$$ $$-0.41231$$ $$\Gamma_0(N)$$-optimal
585.h2 585h2 $$[1, -1, 0, 36, -27]$$ $$6967871/4225$$ $$-3080025$$ $$$$ $$96$$ $$-0.065735$$

## Rank

sage: E.rank()

The elliptic curves in class 585.h have rank $$1$$.

## Complex multiplication

The elliptic curves in class 585.h do not have complex multiplication.

## Modular form585.2.a.h

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + q^{5} - 4 q^{7} - 3 q^{8} + q^{10} - 2 q^{11} - q^{13} - 4 q^{14} - q^{16} - 2 q^{17} - 6 q^{19} + O(q^{20})$$

## 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. 