Properties

Label 2.19.ak_cl
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:  $( 1 - 5 x + 19 x^{2} )^{2}$
Frobenius angles:  $\pm0.305569972467$, $\pm0.305569972467$
Angle rank:  $1$ (numerical)
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 225 140625 49280400 17128265625 6129709430625 2212197156000000 798909821626178025 288439879410444515625 104127868974500422131600 37590033254766349306640625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 388 7180 131428 2475550 47022118 893763370 16983472708 322689305140 6131076010948

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The isogeny class factors as 1.19.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
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.a_n$2$(not in LMFDB)
2.19.k_cl$2$(not in LMFDB)
2.19.f_g$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.19.a_n$2$(not in LMFDB)
2.19.k_cl$2$(not in LMFDB)
2.19.f_g$3$(not in LMFDB)
2.19.a_an$4$(not in LMFDB)
2.19.af_g$6$(not in LMFDB)