# Properties

 Label 189618.p Number of curves $4$ Conductor $189618$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 189618.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
189618.p1 189618ba3 $$[1, 0, 1, -2804481996, -57164824360694]$$ $$505384091400037554067434625/815656731648$$ $$3937019253229151232$$ $$[2]$$ $$74649600$$ $$3.7234$$
189618.p2 189618ba4 $$[1, 0, 1, -2804454956, -57165981802486]$$ $$-505369473241574671219626625/20303219722982711328$$ $$-97999763687870457882392352$$ $$[2]$$ $$149299200$$ $$4.0700$$
189618.p3 189618ba1 $$[1, 0, 1, -34720716, -77954209526]$$ $$959024269496848362625/11151660319506432$$ $$53826934395136521535488$$ $$[2]$$ $$24883200$$ $$3.1741$$ $$\Gamma_0(N)$$-optimal
189618.p4 189618ba2 $$[1, 0, 1, -7031756, -198855284470]$$ $$-7966267523043306625/3534510366354604032$$ $$-17060406446913699933093888$$ $$[2]$$ $$49766400$$ $$3.5207$$

## Rank

sage: E.rank()

The elliptic curves in class 189618.p have rank $$0$$.

## Complex multiplication

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

## Modular form 189618.2.a.p

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.