# Properties

 Label 5586.y Number of curves $4$ Conductor $5586$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("y1")

sage: E.isogeny_class()

## Elliptic curves in class 5586.y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5586.y1 5586bb3 $$[1, 0, 0, -4290049, 3419763689]$$ $$74220219816682217473/16416$$ $$1931325984$$ $$$$ $$92160$$ $$2.0744$$
5586.y2 5586bb2 $$[1, 0, 0, -268129, 53416649]$$ $$18120364883707393/269485056$$ $$31704647353344$$ $$[2, 2]$$ $$46080$$ $$1.7278$$
5586.y3 5586bb4 $$[1, 0, 0, -260289, 56689065]$$ $$-16576888679672833/2216253521952$$ $$-260740010604130848$$ $$$$ $$92160$$ $$2.0744$$
5586.y4 5586bb1 $$[1, 0, 0, -17249, 782025]$$ $$4824238966273/537919488$$ $$63285689843712$$ $$$$ $$23040$$ $$1.3812$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form5586.2.a.y

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} + q^{8} + q^{9} - 2q^{10} - 4q^{11} + q^{12} - 2q^{13} - 2q^{15} + q^{16} + 6q^{17} + q^{18} + q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 