Properties

Label 2.19.aj_bu
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:  $( 1 - 8 x + 19 x^{2} )( 1 - x + 19 x^{2} )$
Frobenius angles:  $\pm0.130073469147$, $\pm0.463406802480$
Angle rank:  $1$ (numerical)
Jacobians:  10

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 10 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 228 134064 47056464 16905470400 6130377927948 2214310804183296 799096559016987228 288443121410865657600 104127350297602681851984 37590003660735378639373104

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 11 373 6860 129721 2475821 47067046 893972279 16983663601 322687697780 6131071184053

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The isogeny class factors as 1.19.ai $\times$ 1.19.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{19}$
The base change of $A$ to $\F_{19^{6}}$ is 1.47045881.pra 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
All geometric endomorphisms are defined over $\F_{19^{6}}$.
Remainder of endomorphism lattice by field

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.ah_be$2$(not in LMFDB)
2.19.h_be$2$(not in LMFDB)
2.19.ap_dq$3$(not in LMFDB)
2.19.ag_bf$3$(not in LMFDB)
2.19.a_aba$3$(not in LMFDB)
2.19.a_al$3$(not in LMFDB)
2.19.a_bl$3$(not in LMFDB)
2.19.g_bf$3$(not in LMFDB)
2.19.j_bu$3$(not in LMFDB)
2.19.p_dq$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.19.ah_be$2$(not in LMFDB)
2.19.h_be$2$(not in LMFDB)
2.19.ap_dq$3$(not in LMFDB)
2.19.ag_bf$3$(not in LMFDB)
2.19.a_aba$3$(not in LMFDB)
2.19.a_al$3$(not in LMFDB)
2.19.a_bl$3$(not in LMFDB)
2.19.g_bf$3$(not in LMFDB)
2.19.j_bu$3$(not in LMFDB)
2.19.p_dq$3$(not in LMFDB)
2.19.aq_dy$6$(not in LMFDB)
2.19.ao_dj$6$(not in LMFDB)
2.19.ai_bt$6$(not in LMFDB)
2.19.ac_bn$6$(not in LMFDB)
2.19.ab_as$6$(not in LMFDB)
2.19.a_al$6$(not in LMFDB)
2.19.b_as$6$(not in LMFDB)
2.19.c_bn$6$(not in LMFDB)
2.19.i_bt$6$(not in LMFDB)
2.19.o_dj$6$(not in LMFDB)
2.19.q_dy$6$(not in LMFDB)
2.19.a_abl$12$(not in LMFDB)
2.19.a_l$12$(not in LMFDB)
2.19.a_ba$12$(not in LMFDB)