Properties

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

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Invariants

Base field:  $\F_{19}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x + 19 x^{2} )( 1 - 3 x + 19 x^{2} )$
Frobenius angles:  $\pm0.203259864187$, $\pm0.388176076177$
Angle rank:  $2$ (numerical)
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 221 137241 48439664 17046567369 6131185293701 2213482610416896 799053232013574941 288443891852288625609 104127042519008111932784 37589914962904844325667401

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 380 7060 130804 2476150 47049446 893923810 16983708964 322686743980 6131056717100

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The isogeny class factors as 1.19.ah $\times$ 1.19.ad 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_r$2$(not in LMFDB)
2.19.e_r$2$(not in LMFDB)
2.19.k_ch$2$(not in LMFDB)
2.19.ae_bp$3$(not in LMFDB)
2.19.f_o$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.19.ae_r$2$(not in LMFDB)
2.19.e_r$2$(not in LMFDB)
2.19.k_ch$2$(not in LMFDB)
2.19.ae_bp$3$(not in LMFDB)
2.19.f_o$3$(not in LMFDB)
2.19.al_ck$6$(not in LMFDB)
2.19.af_o$6$(not in LMFDB)
2.19.ac_bj$6$(not in LMFDB)
2.19.c_bj$6$(not in LMFDB)
2.19.e_bp$6$(not in LMFDB)
2.19.l_ck$6$(not in LMFDB)