Abelian variety isogeny class 3.13.av_he_abif over $\F_{13}$

• Label: 3.13.av_he_abif
• Base field: $\F_{13}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{13}$
Dimension:	$3$
L-polynomial:	$( 1 - 7 x + 13 x^{2} )^{3}$
	$1 - 21 x + 186 x^{2} - 889 x^{3} + 2418 x^{4} - 3549 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.0772104791556$, $\pm0.0772104791556$, $\pm0.0772104791556$
Angle rank:	$1$ (numerical)

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$343$	$3176523$	$9636401152$	$22836204908811$	$51009911926504363$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$101$	$1988$	$27989$	$370013$	$4825292$	$62754545$	$815792645$	$10604854484$	$137860172621$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ah^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.13.ah_ak_gf	$2$	(not in LMFDB)
3.13.h_ak_agf	$2$	(not in LMFDB)
3.13.v_he_bif	$2$	(not in LMFDB)
3.13.am_ci_aig	$3$	(not in LMFDB)
3.13.aj_s_l	$3$	(not in LMFDB)
3.13.ad_p_aec	$3$	(not in LMFDB)
3.13.a_a_acs	$3$	(not in LMFDB)
3.13.d_ag_adt	$3$	(not in LMFDB)
3.13.g_bz_gi	$3$	(not in LMFDB)
3.13.j_cl_ju	$3$	(not in LMFDB)
3.13.m_dg_ny	$3$	(not in LMFDB)
3.13.p_ek_tv	$3$	(not in LMFDB)