# Properties

 Label 116688k Number of curves $2$ Conductor $116688$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 116688k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
116688.a1 116688k1 $$[0, -1, 0, -631320, -165133584]$$ $$6793805286030262681/1048227429629952$$ $$4293539551764283392$$ $$$$ $$3612672$$ $$2.2987$$ $$\Gamma_0(N)$$-optimal
116688.a2 116688k2 $$[0, -1, 0, 1099240, -912735504]$$ $$35862531227445945959/108547797844556928$$ $$-444611779971305177088$$ $$$$ $$7225344$$ $$2.6453$$

## Rank

sage: E.rank()

The elliptic curves in class 116688k have rank $$0$$.

## Complex multiplication

The elliptic curves in class 116688k do not have complex multiplication.

## Modular form 116688.2.a.k

sage: E.q_eigenform(10)

$$q - q^{3} - 4q^{5} + 2q^{7} + q^{9} - q^{11} - q^{13} + 4q^{15} - q^{17} + 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 