# Properties

 Label 2.19.an_cz 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: $1 - 13 x + 77 x^{2} - 247 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.0986133210333$, $\pm0.318874605641$ Angle rank: $2$ (numerical) Number field: 4.0.64389.1 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:

• $y^2=8x^6+10x^5+7x^4+16x^3+9x^2+12x+10$
• $y^2=15x^6+6x^5+11x^4+x^3+18x^2+x+15$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 179 125121 47489237 17014078701 6128624886224 2212876729285569 798996913370243249 288445608159578024949 104128030656023260494527 37590028970091756735864576

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 347 6925 130555 2475112 47036567 893860807 16983810019 322689806185 6131075312102

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.64389.1.
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.n_cz $2$ (not in LMFDB)