Abelian variety isogeny class 3.13.a_a_t over $\F_{13}$

• Label: 3.13.a_a_t
• Base field: $\F_{13}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{13}$
Dimension:	$3$
L-polynomial:	$1 + 19 x^{3} + 2197 x^{6}$
Frobenius angles:	$\pm0.188321590267$, $\pm0.478345076400$, $\pm0.854988256933$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{9})\)
Galois group:	$C_6$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2217$	$4830843$	$10896752313$	$23298078511011$	$51185893399767597$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$170$	$2255$	$28562$	$371294$	$4838909$	$62748518$	$815730722$	$10604144264$	$137858491850$

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 $\F_{13^{3}}$.

Endomorphism algebra over $\F_{13}$

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

Endomorphism algebra over $\overline{\F}_{13}$

The base change of $A$ to $\F_{13^{3}}$ is 1.2197.t^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
3.13.a_a_at	$2$	(not in LMFDB)
3.13.ap_ek_atv	$9$	(not in LMFDB)
3.13.am_dg_any	$9$	(not in LMFDB)