Abelian variety isogeny class 3.7.an_cz_akc over $\F_{7}$

• Label: 3.7.an_cz_akc
• Base field: $\F_{7}$
• 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:	$\F_{7}$
Dimension:	$3$
L-polynomial:	$( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 7 x^{2} )^{2}$
	$1 - 13 x + 77 x^{2} - 262 x^{3} + 539 x^{4} - 637 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.227185525829$, $\pm0.227185525829$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$48$	$89856$	$42928704$	$14821208064$	$4889951756688$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$35$	$364$	$2567$	$17305$	$118508$	$823783$	$5761007$	$40341028$	$282464075$

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 $\F_{7^{6}}$.

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.af $\times$ 1.7.ae^2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.7.af : \(\Q(\sqrt{-3}) \).
• 1.7.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Endomorphism algebra over $\overline{\F}_{7}$

The base change of $A$ to $\F_{7^{6}}$ is 1.117649.la^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{7^{2}}$
  The base change of $A$ to $\F_{7^{2}}$ is 1.49.al $\times$ 1.49.ac^2. The endomorphism algebra for each factor is:

  • 1.49.al : \(\Q(\sqrt{-3}) \).
  • 1.49.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• Endomorphism algebra over $\F_{7^{3}}$
  The base change of $A$ to $\F_{7^{3}}$ is 1.343.au $\times$ 1.343.u^2. The endomorphism algebra for each factor is:

  • 1.343.au : \(\Q(\sqrt{-3}) \).
  • 1.343.u^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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
3.7.af_f_k	$2$	(not in LMFDB)
3.7.ad_ad_bm	$2$	(not in LMFDB)
3.7.d_ad_abm	$2$	(not in LMFDB)
3.7.f_f_ak	$2$	(not in LMFDB)
3.7.n_cz_kc	$2$	(not in LMFDB)
3.7.ak_by_age	$3$	(not in LMFDB)
3.7.ah_bd_ade	$3$	(not in LMFDB)
3.7.ah_bg_adz	$3$	(not in LMFDB)
3.7.ae_ae_bs	$3$	(not in LMFDB)
3.7.ae_f_i	$3$	(not in LMFDB)
3.7.ae_u_aca	$3$	(not in LMFDB)
3.7.ab_ae_l	$3$	(not in LMFDB)
3.7.ab_f_c	$3$	(not in LMFDB)
3.7.ab_u_an	$3$	(not in LMFDB)
3.7.c_c_i	$3$	(not in LMFDB)
3.7.c_o_bg	$3$	(not in LMFDB)
3.7.f_ae_acd	$3$	(not in LMFDB)
3.7.f_f_ak	$3$	(not in LMFDB)
3.7.f_u_cn	$3$	(not in LMFDB)
3.7.i_bg_do	$3$	(not in LMFDB)
3.7.l_ce_gx	$3$	(not in LMFDB)
3.7.o_di_lk	$3$	(not in LMFDB)