# Properties

 Label 1638.h Number of curves $2$ Conductor $1638$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 1638.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.h1 1638h2 $$[1, -1, 0, -33070464, 73207840986]$$ $$-5486773802537974663600129/2635437714$$ $$-1921234093506$$ $$[]$$ $$65856$$ $$2.5959$$
1638.h2 1638h1 $$[1, -1, 0, 6426, 2238516]$$ $$40251338884511/2997011332224$$ $$-2184821261191296$$ $$[]$$ $$9408$$ $$1.6229$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1638.h have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1638.h do not have complex multiplication.

## Modular form1638.2.a.h

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 