# Properties

 Label 116886p Number of curves $4$ Conductor $116886$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 116886p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
116886.w3 116886p1 $$[1, 0, 1, -15975, -718742]$$ $$254478514753/21762048$$ $$38552795516928$$ $$$$ $$414720$$ $$1.3481$$ $$\Gamma_0(N)$$-optimal
116886.w2 116886p2 $$[1, 0, 1, -54695, 4098026]$$ $$10214075575873/1806590016$$ $$3200484415334976$$ $$[2, 2]$$ $$829440$$ $$1.6947$$
116886.w4 116886p3 $$[1, 0, 1, 105025, 23583866]$$ $$72318867421247/177381135624$$ $$-314241502007189064$$ $$$$ $$1658880$$ $$2.0413$$
116886.w1 116886p4 $$[1, 0, 1, -833935, 293040218]$$ $$36204575259448513/1527466248$$ $$2705999633773128$$ $$$$ $$1658880$$ $$2.0413$$

## Rank

sage: E.rank()

The elliptic curves in class 116886p have rank $$0$$.

## Complex multiplication

The elliptic curves in class 116886p do not have complex multiplication.

## Modular form 116886.2.a.p

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - 2q^{10} + q^{12} - 2q^{13} + q^{14} + 2q^{15} + q^{16} - 2q^{17} - q^{18} + O(q^{20})$$ ## 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. 