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

• Label: 3.7.ak_bu_afk
• Base field: $\F_{7}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• 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:	$( 1 + 7 x^{2} )( 1 - 5 x + 7 x^{2} )^{2}$
	$1 - 10 x + 46 x^{2} - 140 x^{3} + 322 x^{4} - 490 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.106147807505$, $\pm0.5$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$72$	$97344$	$36111744$	$13039812864$	$4762544934312$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$42$	$304$	$2258$	$16858$	$118908$	$826054$	$5768546$	$40378768$	$282574842$

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 $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.af^2 $\times$ 1.7.a 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.a : \(\Q(\sqrt{-7}) \).

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

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

• 1.49.al^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 1.49.o : the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$.

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.a_ae_a	$2$	(not in LMFDB)
3.7.k_bu_fk	$2$	(not in LMFDB)
3.7.ae_q_ace	$3$	(not in LMFDB)
3.7.ab_b_ao	$3$	(not in LMFDB)
3.7.c_w_bc	$3$	(not in LMFDB)
3.7.f_z_cs	$3$	(not in LMFDB)
3.7.i_bl_ei	$3$	(not in LMFDB)