Properties

Label 2.19.al_ck
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:  $( 1 - 8 x + 19 x^{2} )( 1 - 3 x + 19 x^{2} )$
Frobenius angles:  $\pm0.130073469147$, $\pm0.388176076177$
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
$A(\F_{q^r})$ 204 131376 47655216 16974304704 6126953978724 2213482610416896 799091751465814164 288449990483531192064 104127672854620178665776 37589962955782452689708976

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 9 365 6948 130249 2474439 47049446 893966901 16984068049 322688697372 6131064544925

Decomposition and endomorphism algebra

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