# Properties

 Label 2.19.aj_bp Base Field $\F_{19}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{19}$ Dimension: $2$ L-polynomial: $1 - 9 x + 41 x^{2} - 171 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.0387402455333$, $\pm0.487338161656$ Angle rank: $2$ (numerical) Number field: 4.0.404685.1 Galois group: $D_{4}$ Jacobians: 2

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

• $y^2=6x^6+7x^5+12x^4+11x^3+9x^2+13x+14$
• $y^2=2x^6+7x^5+9x^4+7x^3+2x^2+14x+15$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 223 130009 46130449 16808213565 6123139391248 2213461142579401 798986180611862053 288434720228854856085 104127054027894241747699 37589990931555184078299904

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 11 363 6725 128971 2472896 47048991 893848799 16983168931 322686779645 6131069107878

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.404685.1.
All geometric endomorphisms are defined over $\F_{19}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.j_bp $2$ (not in LMFDB)