# Properties

 Label 116160cd Number of curves $4$ Conductor $116160$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cd1")

sage: E.isogeny_class()

## Elliptic curves in class 116160cd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
116160.dc4 116160cd1 $$[0, -1, 0, 105835, 48657237]$$ $$72268906496/606436875$$ $$-1100124074712960000$$ $$[2]$$ $$1105920$$ $$2.1432$$ $$\Gamma_0(N)$$-optimal
116160.dc3 116160cd2 $$[0, -1, 0, -1527665, 669713937]$$ $$13584145739344/1195803675$$ $$34708507103832883200$$ $$[2]$$ $$2211840$$ $$2.4898$$
116160.dc2 116160cd3 $$[0, -1, 0, -7560725, 8010763125]$$ $$-26348629355659264/24169921875$$ $$-43846134750000000000$$ $$[2]$$ $$3317760$$ $$2.6925$$
116160.dc1 116160cd4 $$[0, -1, 0, -120998225, 512331200625]$$ $$6749703004355978704/5671875$$ $$164627620608000000$$ $$[2]$$ $$6635520$$ $$3.0391$$

## Rank

sage: E.rank()

The elliptic curves in class 116160cd have rank $$1$$.

## Complex multiplication

The elliptic curves in class 116160cd do not have complex multiplication.

## Modular form 116160.2.a.cd

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} - 2q^{7} + q^{9} + 2q^{13} - q^{15} + 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.