# Properties

 Label 190740u Number of curves $4$ Conductor $190740$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("u1")

sage: E.isogeny_class()

## Elliptic curves in class 190740u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
190740.k4 190740u1 $$[0, -1, 0, 63195, -22487850]$$ $$72268906496/606436875$$ $$-234206590631310000$$ $$[2]$$ $$1492992$$ $$2.0143$$ $$\Gamma_0(N)$$-optimal
190740.k3 190740u2 $$[0, -1, 0, -912180, -308467800]$$ $$13584145739344/1195803675$$ $$7389131191236115200$$ $$[2]$$ $$2985984$$ $$2.3609$$
190740.k2 190740u3 $$[0, -1, 0, -4514565, -3693508038]$$ $$-26348629355659264/24169921875$$ $$-9334450511718750000$$ $$[2]$$ $$4478976$$ $$2.5636$$
190740.k1 190740u4 $$[0, -1, 0, -72248940, -236347539288]$$ $$6749703004355978704/5671875$$ $$35047750188000000$$ $$[2]$$ $$8957952$$ $$2.9102$$

## Rank

sage: E.rank()

The elliptic curves in class 190740u have rank $$1$$.

## Complex multiplication

The elliptic curves in class 190740u do not have complex multiplication.

## Modular form 190740.2.a.u

sage: E.q_eigenform(10)

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

## 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.