# Properties

 Label 2.19.al_cn 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: $1 - 11 x + 65 x^{2} - 209 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.183908064930$, $\pm0.360590248890$ Angle rank: $2$ (numerical) Number field: 4.0.26533.1 Galois group: $D_{4}$ Jacobians: 5

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 5 curves, and hence is principally polarizable:

• $y^2=5x^5+12x^4+15x^3+4x^2+11x+14$
• $y^2=17x^6+17x^5+16x^4+3x^3+9x^2+2x+17$
• $y^2=10x^6+17x^5+2x^4+17x^3+4x+10$
• $y^2=9x^6+x^5+6x^4+6x^3+18x^2+2x+18$
• $y^2=15x^6+x^5+5x^4+8x^3+3x^2+3x+13$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 207 133929 48346713 17064295677 6132261832272 2213364397163049 799043845092737661 288445952175817883733 104127441596475871887171 37589936977009790399856384

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 371 7047 130939 2476584 47046935 893913309 16983830275 322687980711 6131060307686

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.26533.1.
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.l_cn $2$ (not in LMFDB)