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

• Label: 3.7.aj_bj_ads
• 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:	$( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 8 x^{2} - 28 x^{3} + 49 x^{4} )$
	$1 - 9 x + 35 x^{2} - 96 x^{3} + 245 x^{4} - 441 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.0704914820143$, $\pm0.106147807505$, $\pm0.570491482014$
Angle rank:	$2$ (numerical)
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$78$	$91260$	$32760936$	$13026452400$	$4787838907098$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$39$	$272$	$2255$	$16949$	$117936$	$822947$	$5770991$	$40382984$	$282508239$

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 $\F_{7^{4}}$.

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.af $\times$ 2.7.ae_i 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 : \(\Q(\sqrt{-3}) \).
• 2.7.ae_i : \(\Q(i, \sqrt{10})\).

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

The base change of $A$ to $\F_{7^{4}}$ is 1.2401.ack^2 $\times$ 1.2401.ax. The endomorphism algebra for each factor is:

• 1.2401.ack^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$
• 1.2401.ax : \(\Q(\sqrt{-3}) \).

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 $\times$ 2.49.a_ack. The endomorphism algebra for each factor is:

  • 1.49.al : \(\Q(\sqrt{-3}) \).
  • 2.49.a_ack : \(\Q(i, \sqrt{10})\).

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.ab_af_q	$2$	(not in LMFDB)
3.7.b_af_aq	$2$	(not in LMFDB)
3.7.j_bj_ds	$2$	(not in LMFDB)
3.7.ad_l_abw	$3$	(not in LMFDB)
3.7.a_ab_ay	$3$	(not in LMFDB)