Properties

Label 2.19.ak_cc
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:  $( 1 - 8 x + 19 x^{2} )( 1 - 2 x + 19 x^{2} )$
Frobenius angles:  $\pm0.130073469147$, $\pm0.426318466621$
Angle rank:  $2$ (numerical)
Jacobians:  16

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 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 216 133056 47396664 16933238784 6127476737976 2213929646824896 799111425953312856 288447031986733154304 104127345957440010149784 37589970685374618644016576

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 370 6910 129934 2474650 47058946 893988910 16983893854 322687684330 6131065805650

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The isogeny class factors as 1.19.ai $\times$ 1.19.ac 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.ag_w$2$(not in LMFDB)
2.19.g_w$2$(not in LMFDB)
2.19.k_cc$2$(not in LMFDB)
2.19.ab_bk$3$(not in LMFDB)
2.19.f_y$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.19.ag_w$2$(not in LMFDB)
2.19.g_w$2$(not in LMFDB)
2.19.k_cc$2$(not in LMFDB)
2.19.ab_bk$3$(not in LMFDB)
2.19.f_y$3$(not in LMFDB)
2.19.aj_ca$6$(not in LMFDB)
2.19.af_y$6$(not in LMFDB)
2.19.ad_bo$6$(not in LMFDB)
2.19.b_bk$6$(not in LMFDB)
2.19.d_bo$6$(not in LMFDB)
2.19.j_ca$6$(not in LMFDB)