Abelian variety isogeny class 4.3.a_g_a_bb over $\F_{3}$

• Label: 4.3.a_g_a_bb
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$( 1 + 3 x^{2} + 9 x^{4} )^{2}$
	$1 + 6 x^{2} + 27 x^{4} + 54 x^{6} + 81 x^{8}$
Frobenius angles:	$\pm0.333333333333$, $\pm0.333333333333$, $\pm0.666666666667$, $\pm0.666666666667$
Angle rank:	$0$ (numerical)
Jacobians:	$4$
Cyclic group of points:	no
Non-cyclic primes:	$13$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$169$	$28561$	$456976$	$68574961$	$3515659849$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$22$	$28$	$118$	$244$	$514$	$2188$	$6886$	$19684$	$60022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):

• $x y+t^2=y^2 z+z^3+x^2 t+y^2 t=0$
• $x y+t^2=y^2 z-z^3+x^2 t+y^2 t=0$
• $x^2+y^2+z t=y^3+y^2 z+z^3+x z t+y z t+x t^2-y t^2+z t^2-t^3=0$
• $x^2+y^2+z t=y^3+y^2 z+z^3+x z t-y z t+z^2 t-x t^2-y t^2-z t^2=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 2.3.a_d^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{12})\)$)$

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{6}}$ is 1.729.acc^4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$

  The base change of $A$ to $\F_{3^{2}}$ is 1.9.d^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{3^{3}}$

  The base change of $A$ to $\F_{3^{3}}$ is 2.27.a_acc^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q(\sqrt{3}) \) ramified at both real infinite places.

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
4.3.a_am_a_cc	$3$	(not in LMFDB)
4.3.a_ad_a_a	$3$	(not in LMFDB)
4.3.am_co_aii_rr	$4$	(not in LMFDB)