Properties

Label 3.16.ay_jg_abxg
Base Field $\F_{2^{4}}$
Dimension $3$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x )^{6}$
Frobenius angles:  $0$, $0$, $0$, $0$, $0$, $0$
Angle rank:  $0$ (numerical)

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 729 11390625 62523502209 274941996890625 1146182576381093889 4715453174592516890625 19335730644885715992608769 79220909236042181489028890625 324511126089026151457611904974849 1329220389899448367763203918212890625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -7 161 3713 64001 1042433 16752641 268337153 4294574081 68717903873 1099505336321

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.

SubfieldPrimitive Model
$\F_{2}$3.2.a_ac_a
$\F_{2}$3.2.a_g_a

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ai_aq_jw$2$(not in LMFDB)
3.16.i_aq_ajw$2$(not in LMFDB)
3.16.y_jg_bxg$2$(not in LMFDB)
3.16.am_bw_aey$3$(not in LMFDB)
3.16.a_a_aey$3$(not in LMFDB)
3.16.m_ds_rg$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ai_aq_jw$2$(not in LMFDB)
3.16.i_aq_ajw$2$(not in LMFDB)
3.16.y_jg_bxg$2$(not in LMFDB)
3.16.am_bw_aey$3$(not in LMFDB)
3.16.a_a_aey$3$(not in LMFDB)
3.16.m_ds_rg$3$(not in LMFDB)
3.16.aq_ei_ats$4$(not in LMFDB)
3.16.ai_bw_ajw$4$(not in LMFDB)
3.16.a_aq_a$4$(not in LMFDB)
3.16.a_bw_a$4$(not in LMFDB)
3.16.i_bw_jw$4$(not in LMFDB)
3.16.q_ei_ts$4$(not in LMFDB)
3.16.ae_a_a$5$(not in LMFDB)
3.16.au_gu_abim$6$(not in LMFDB)
3.16.aq_ey_ayq$6$(not in LMFDB)
3.16.am_ds_arg$6$(not in LMFDB)
3.16.ai_bg_aey$6$(not in LMFDB)
3.16.ae_aq_ey$6$(not in LMFDB)
3.16.ae_bg_acm$6$(not in LMFDB)
3.16.a_a_ey$6$(not in LMFDB)
3.16.e_aq_aey$6$(not in LMFDB)
3.16.e_bg_cm$6$(not in LMFDB)
3.16.i_bg_ey$6$(not in LMFDB)
3.16.m_bw_ey$6$(not in LMFDB)
3.16.q_ey_yq$6$(not in LMFDB)
3.16.u_gu_bim$6$(not in LMFDB)
3.16.ai_q_a$8$(not in LMFDB)
3.16.a_q_a$8$(not in LMFDB)
3.16.i_q_a$8$(not in LMFDB)
3.16.a_a_cm$9$(not in LMFDB)
3.16.am_cm_ajw$10$(not in LMFDB)
3.16.e_a_a$10$(not in LMFDB)
3.16.m_cm_jw$10$(not in LMFDB)
3.16.am_dc_aou$12$(not in LMFDB)
3.16.ai_a_ey$12$(not in LMFDB)
3.16.ai_cm_ajw$12$(not in LMFDB)
3.16.ae_a_cm$12$(not in LMFDB)
3.16.ae_q_aey$12$(not in LMFDB)
3.16.ae_bw_aey$12$(not in LMFDB)
3.16.a_a_a$12$(not in LMFDB)
3.16.a_bg_a$12$(not in LMFDB)
3.16.e_a_acm$12$(not in LMFDB)
3.16.e_q_ey$12$(not in LMFDB)
3.16.e_bw_ey$12$(not in LMFDB)
3.16.i_a_aey$12$(not in LMFDB)
3.16.i_cm_jw$12$(not in LMFDB)
3.16.m_dc_ou$12$(not in LMFDB)
3.16.i_bw_hk$15$(not in LMFDB)
3.16.a_a_acm$18$(not in LMFDB)
3.16.ae_bg_aey$20$(not in LMFDB)
3.16.e_bg_ey$20$(not in LMFDB)
3.16.ae_q_a$24$(not in LMFDB)
3.16.e_q_a$24$(not in LMFDB)
3.16.ai_bw_ahk$30$(not in LMFDB)
3.16.a_q_acm$30$(not in LMFDB)
3.16.a_q_cm$30$(not in LMFDB)