Properties

Label 2.19.ao_di
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:  $( 1 - 8 x + 19 x^{2} )( 1 - 6 x + 19 x^{2} )$
Frobenius angles:  $\pm0.130073469147$, $\pm0.258380448083$
Angle rank:  $2$ (numerical)
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 168 122304 47532744 17083422720 6139358981448 2213711304831936 799013955490370088 288441475423798394880 104127474329448490192104 37589999149034860983559104

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 338 6930 131086 2479446 47054306 893879874 16983566686 322688082150 6131070448178

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The isogeny class factors as 1.19.ai $\times$ 1.19.ag 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.ac_ak$2$(not in LMFDB)
2.19.c_ak$2$(not in LMFDB)
2.19.o_di$2$(not in LMFDB)
2.19.af_bg$3$(not in LMFDB)
2.19.b_ae$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.19.ac_ak$2$(not in LMFDB)
2.19.c_ak$2$(not in LMFDB)
2.19.o_di$2$(not in LMFDB)
2.19.af_bg$3$(not in LMFDB)
2.19.b_ae$3$(not in LMFDB)
2.19.an_dc$6$(not in LMFDB)
2.19.ah_bs$6$(not in LMFDB)
2.19.ab_ae$6$(not in LMFDB)
2.19.f_bg$6$(not in LMFDB)
2.19.h_bs$6$(not in LMFDB)
2.19.n_dc$6$(not in LMFDB)