# Properties

 Label 189618x Number of curves $4$ Conductor $189618$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 189618x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
189618.n4 189618x1 $$[1, 0, 1, -988147, -295745266]$$ $$22106889268753393/4969545596928$$ $$23987047413162442752$$ $$$$ $$6193152$$ $$2.4316$$ $$\Gamma_0(N)$$-optimal
189618.n2 189618x2 $$[1, 0, 1, -14832627, -21987276530]$$ $$74768347616680342513/5615307472896$$ $$27104016647941668864$$ $$[2, 2]$$ $$12386304$$ $$2.7782$$
189618.n3 189618x3 $$[1, 0, 1, -13859187, -24997542386]$$ $$-60992553706117024753/20624795251201152$$ $$-99551947341654981283968$$ $$$$ $$24772608$$ $$3.1247$$
189618.n1 189618x4 $$[1, 0, 1, -237317747, -1407179633650]$$ $$306234591284035366263793/1727485056$$ $$8338240415666304$$ $$$$ $$24772608$$ $$3.1247$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 189618.2.a.x

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 Cremona 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 Cremona labels. 