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

• Label: 3.7.ao_di_alk
• 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:	$( 1 - 4 x + 7 x^{2} )( 1 - 5 x + 7 x^{2} )^{2}$
	$1 - 14 x + 86 x^{2} - 296 x^{3} + 602 x^{4} - 686 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.106147807505$, $\pm0.227185525829$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$73008$	$38211264$	$14126463936$	$4829415508116$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$26$	$324$	$2450$	$17094$	$118508$	$825546$	$5769314$	$40366188$	$282519146$

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

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^2 $\times$ 1.49.ac. The endomorphism algebra for each factor is:

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

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

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.ag_g_q	$2$	(not in LMFDB)
3.7.ae_ae_bs	$2$	(not in LMFDB)
3.7.e_ae_abs	$2$	(not in LMFDB)
3.7.g_g_aq	$2$	(not in LMFDB)
3.7.o_di_lk	$2$	(not in LMFDB)
3.7.al_ce_agx	$3$	(not in LMFDB)
3.7.ai_bg_ado	$3$	(not in LMFDB)
3.7.af_ae_cd	$3$	(not in LMFDB)
3.7.af_f_k	$3$	(not in LMFDB)
3.7.af_u_acn	$3$	(not in LMFDB)
3.7.ac_c_ai	$3$	(not in LMFDB)
3.7.ac_o_abg	$3$	(not in LMFDB)
3.7.b_ae_al	$3$	(not in LMFDB)
3.7.b_f_ac	$3$	(not in LMFDB)
3.7.b_u_n	$3$	(not in LMFDB)
3.7.e_ae_abs	$3$	(not in LMFDB)
3.7.e_f_ai	$3$	(not in LMFDB)
3.7.e_u_ca	$3$	(not in LMFDB)
3.7.h_bd_de	$3$	(not in LMFDB)
3.7.h_bg_dz	$3$	(not in LMFDB)
3.7.k_by_ge	$3$	(not in LMFDB)
3.7.n_cz_kc	$3$	(not in LMFDB)