# Properties

 Label 2.19.an_db 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: $1 - 13 x + 79 x^{2} - 247 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.161616782251$, $\pm0.288204697885$ Angle rank: $2$ (numerical) Number field: 4.0.21125.1 Galois group: $C_4$ 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:

• $y^2=18x^6+2x^5+16x^4+2x^3+11x^2+13x+3$
• $y^2=15x^6+5x^5+18x^4+10x^3+11x^2+15$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 181 126881 48039391 17109268445 6138926902096 2213531724084521 799002280193961451 288441556256912702645 104127508988997935788441 37589990922529927848066816

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 351 7003 131283 2479272 47050491 893866813 16983571443 322688189557 6131069106406

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.21125.1.
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_db $2$ (not in LMFDB)