# Properties

 Label 5586.p Number of curves $2$ Conductor $5586$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 5586.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5586.p1 5586r1 $$[1, 0, 1, -4681, 122840]$$ $$96386901625/18468$$ $$2172741732$$ $$$$ $$5760$$ $$0.79226$$ $$\Gamma_0(N)$$-optimal
5586.p2 5586r2 $$[1, 0, 1, -4191, 149692]$$ $$-69173457625/42633378$$ $$-5015774288322$$ $$$$ $$11520$$ $$1.1388$$

## Rank

sage: E.rank()

The elliptic curves in class 5586.p have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5586.p do not have complex multiplication.

## Modular form5586.2.a.p

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + 4q^{11} + q^{12} + q^{16} + 2q^{17} - q^{18} - q^{19} + O(q^{20})$$ ## 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. 