# Properties

 Label 2.19.am_cs 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: $( 1 - 8 x + 19 x^{2} )( 1 - 4 x + 19 x^{2} )$ Frobenius angles: $\pm0.130073469147$, $\pm0.348268167089$ Angle rank: $2$ (numerical) Jacobians: 12

This isogeny class is not 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 12 curves, and hence is principally polarizable:

• $y^2=12x^6+11x^5+9x^4+15x^3+9x^2+11x+12$
• $y^2=15x^6+2x^5+15x^4+16x^3+2x^2+11x+5$
• $y^2=11x^6+3x^5+x^4+18x^3+17x^2+6$
• $y^2=x^6+4x^5+10x^4+12x^3+10x^2+4x+1$
• $y^2=17x^6+15x^5+7x^4+13x^3+9x^2+8x+6$
• $y^2=17x^6+16x^5+11x^4+17x^3+17x^2+17x+8$
• $y^2=10x^6+10x^5+15x^4+6x^3+10x^2+15x+10$
• $y^2=11x^5+3x^4+5x^3+3x^2+11x$
• $y^2=15x^6+2x^5+7x^4+17x^3+7x^2+2x+15$
• $y^2=7x^6+13x^5+11x^4+9x^3+x^2+10x+7$
• $y^2=8x^6+11x^5+16x^4+11x^3+4x^2+9x+12$
• $y^2=8x^6+4x^5+5x^4+8x^3+6x^2+5x+12$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 192 129024 47791296 17020846080 6129255605952 2213192742598656 799047672711483072 288449268959839518720 104127999873776666133696 37589992898409852999727104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 358 6968 130606 2475368 47043286 893917592 16984025566 322689710792 6131069428678

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ai $\times$ 1.19.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{19}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.ae_g $2$ (not in LMFDB) 2.19.e_g $2$ (not in LMFDB) 2.19.m_cs $2$ (not in LMFDB) 2.19.ad_bi $3$ (not in LMFDB) 2.19.d_k $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.ae_g $2$ (not in LMFDB) 2.19.e_g $2$ (not in LMFDB) 2.19.m_cs $2$ (not in LMFDB) 2.19.ad_bi $3$ (not in LMFDB) 2.19.d_k $3$ (not in LMFDB) 2.19.al_co $6$ (not in LMFDB) 2.19.af_bq $6$ (not in LMFDB) 2.19.ad_k $6$ (not in LMFDB) 2.19.d_bi $6$ (not in LMFDB) 2.19.f_bq $6$ (not in LMFDB) 2.19.l_co $6$ (not in LMFDB)