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

• Label: 3.7.a_a_bl
• Base field: $\F_{7}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{7}$
Dimension:	$3$
L-polynomial:	$1 + 37 x^{3} + 343 x^{6}$
Frobenius angles:	$\pm0.328370029727$, $\pm0.338296636940$, $\pm0.995036696394$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{9})\)
Galois group:	$C_6$
Isomorphism classes:	1
Cyclic group of points:	yes

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$381$	$116967$	$55306341$	$13841056011$	$4747565749071$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$50$	$455$	$2402$	$16808$	$115601$	$823544$	$5764802$	$40391348$	$282475250$

Jacobians and polarizations

This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{3}}$.

Endomorphism algebra over $\F_{7}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{9})\).

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

The base change of $A$ to $\F_{7^{3}}$ is 1.343.bl^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
3.7.a_a_abl	$2$	(not in LMFDB)
3.7.ap_ds_amx	$9$	(not in LMFDB)
3.7.aj_bk_adx	$9$	(not in LMFDB)