# Properties

 Label 189618.i Number of curves $4$ Conductor $189618$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 189618.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
189618.i1 189618bs4 $$[1, 1, 0, -424131381339, -106316153461934307]$$ $$1748094148784980747354970849498497/887694600425282263291392$$ $$4284732286584156255995260528128$$ $$[2]$$ $$1588543488$$ $$5.2105$$
189618.i2 189618bs3 $$[1, 1, 0, -58018434139, 2958123786906397]$$ $$4474676144192042711273397261697/1806328356954994499451382272$$ $$8718801970305580044902427012930048$$ $$[2]$$ $$1588543488$$ $$5.2105$$
189618.i3 189618bs2 $$[1, 1, 0, -26651601499, -1642317815705315]$$ $$433744050935826360922067531137/9612122270219882316693504$$ $$46395878282997759945157055348736$$ $$[2, 2]$$ $$794271744$$ $$4.8639$$
189618.i4 189618bs1 $$[1, 1, 0, 151311781, -78662657863395]$$ $$79374649975090937760383/553856914190911653543936$$ $$-2673361538128920087530752180224$$ $$[2]$$ $$397135872$$ $$4.5173$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 189618.i have rank $$1$$.

## Complex multiplication

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

## Modular form 189618.2.a.i

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.