# Properties

 Label 439569n Number of curves $2$ Conductor $439569$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("n1")

sage: E.isogeny_class()

## Elliptic curves in class 439569n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
439569.n2 439569n1 [1, -1, 1, -155681, 682413320] [2] 11698176 $$\Gamma_0(N)$$-optimal*
439569.n1 439569n2 [1, -1, 1, -12610136, 17062512536] [2] 23396352 $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 439569n1.

## Rank

sage: E.rank()

The elliptic curves in class 439569n have rank $$1$$.

## Complex multiplication

The elliptic curves in class 439569n do not have complex multiplication.

## Modular form 439569.2.a.n

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} - 2q^{5} + 3q^{8} + 2q^{10} - 6q^{11} - q^{16} + 2q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.