# Properties

 Label 116688.w Number of curves $2$ Conductor $116688$ CM no Rank $2$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

## Elliptic curves in class 116688.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
116688.w1 116688x1 $$[0, 1, 0, -48328, 3703412]$$ $$3047678972871625/304559880768$$ $$1247477271625728$$ $$$$ $$516096$$ $$1.6330$$ $$\Gamma_0(N)$$-optimal
116688.w2 116688x2 $$[0, 1, 0, 59832, 18023796]$$ $$5783051584712375/37533175779528$$ $$-153735887992946688$$ $$$$ $$1032192$$ $$1.9796$$

## Rank

sage: E.rank()

The elliptic curves in class 116688.w have rank $$2$$.

## Complex multiplication

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

## Modular form 116688.2.a.w

sage: E.q_eigenform(10)

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