# Properties

 Label 189618.a Number of curves $4$ Conductor $189618$ CM no Rank $2$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 189618.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
189618.a1 189618bi4 $$[1, 1, 0, -327356, -66950970]$$ $$803760366578833/65593817586$$ $$316608829068463074$$ $$$$ $$3538944$$ $$2.1007$$
189618.a2 189618bi2 $$[1, 1, 0, -68786, 5707200]$$ $$7457162887153/1370924676$$ $$6617191564438884$$ $$[2, 2]$$ $$1769472$$ $$1.7542$$
189618.a3 189618bi1 $$[1, 1, 0, -65406, 6410916]$$ $$6411014266033/296208$$ $$1429739440272$$ $$$$ $$884736$$ $$1.4076$$ $$\Gamma_0(N)$$-optimal
189618.a4 189618bi3 $$[1, 1, 0, 135704, 33395146]$$ $$57258048889007/132611470002$$ $$-640090236908883618$$ $$$$ $$3538944$$ $$2.1007$$

## Rank

sage: E.rank()

The elliptic curves in class 189618.a have rank $$2$$.

## Complex multiplication

The elliptic curves in class 189618.a do not have complex multiplication.

## Modular form 189618.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - 4q^{7} - q^{8} + q^{9} + 2q^{10} - q^{11} - q^{12} + 4q^{14} + 2q^{15} + q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 