Properties

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

Learn more about

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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)