Properties

Label 2.19.ao_dj
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:  $( 1 - 7 x + 19 x^{2} )^{2}$
Frobenius angles:  $\pm0.203259864187$, $\pm0.203259864187$
Angle rank:  $1$ (numerical)
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 169 123201 47831056 17140831929 6146645853049 2214310804183296 799032354409893409 288437997911468525289 104126719963943210135056 37589913051239921148065601

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 340 6972 131524 2482386 47067046 893900454 16983361924 322685744388 6131056405300

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The isogeny class factors as 1.19.ah 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
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_al$2$(not in LMFDB)
2.19.o_dj$2$(not in LMFDB)
2.19.ai_bt$3$(not in LMFDB)
2.19.ac_bn$3$(not in LMFDB)
2.19.b_as$3$(not in LMFDB)
2.19.h_be$3$(not in LMFDB)
2.19.q_dy$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.19.a_al$2$(not in LMFDB)
2.19.o_dj$2$(not in LMFDB)
2.19.ai_bt$3$(not in LMFDB)
2.19.ac_bn$3$(not in LMFDB)
2.19.b_as$3$(not in LMFDB)
2.19.h_be$3$(not in LMFDB)
2.19.q_dy$3$(not in LMFDB)
2.19.a_l$4$(not in LMFDB)
2.19.aq_dy$6$(not in LMFDB)
2.19.ap_dq$6$(not in LMFDB)
2.19.aj_bu$6$(not in LMFDB)
2.19.ah_be$6$(not in LMFDB)
2.19.ag_bf$6$(not in LMFDB)
2.19.ab_as$6$(not in LMFDB)
2.19.a_aba$6$(not in LMFDB)
2.19.a_bl$6$(not in LMFDB)
2.19.c_bn$6$(not in LMFDB)
2.19.g_bf$6$(not in LMFDB)
2.19.i_bt$6$(not in LMFDB)
2.19.j_bu$6$(not in LMFDB)
2.19.p_dq$6$(not in LMFDB)
2.19.a_abl$12$(not in LMFDB)
2.19.a_ba$12$(not in LMFDB)