Properties

Label 2.19.am_cw
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:  $( 1 - 6 x + 19 x^{2} )^{2}$
Frobenius angles:  $\pm0.258380448083$, $\pm0.258380448083$
Angle rank:  $1$ (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
$A(\F_{q^r})$ 196 132496 48804196 17171481600 6140553384196 2213111967788176 798918529220525476 288432756057522585600 104126968026280916546116 37589989261020791711932816

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 8 366 7112 131758 2479928 47041566 893773112 16983053278 322686513128 6131068835406

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The isogeny class factors as 1.19.ag 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$
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_c$2$(not in LMFDB)
2.19.m_cw$2$(not in LMFDB)
2.19.g_r$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.19.a_c$2$(not in LMFDB)
2.19.m_cw$2$(not in LMFDB)
2.19.g_r$3$(not in LMFDB)
2.19.a_ac$4$(not in LMFDB)
2.19.ag_r$6$(not in LMFDB)