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

• Label: 4.3.ad_d_j_abb
• 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:	$( 1 - 3 x + 3 x^{2} )( 1 + 9 x^{3} + 27 x^{6} )$
	$1 - 3 x + 3 x^{2} + 9 x^{3} - 27 x^{4} + 27 x^{5} + 27 x^{6} - 81 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.277777777778$, $\pm0.388888888889$, $\pm0.944444444444$
Angle rank:	$0$ (numerical)
Jacobians:	$3$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$37$	$4921$	$1418284$	$48427561$	$3886776037$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$7$	$55$	$91$	$271$	$703$	$2269$	$6643$	$19684$	$58807$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which 0 are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{18}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ad $\times$ 3.3.a_a_j and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.3.ad : \(\Q(\sqrt{-3}) \).
• 3.3.a_a_j : \(\Q(\zeta_{9})\).

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

The base change of $A$ to $\F_{3^{18}}$ is 1.387420489.cggc^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.ad $\times$ 3.9.a_a_abb. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 3.9.a_a_abb : \(\Q(\zeta_{9})\).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.j^3. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 1.27.j^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
• Endomorphism algebra over $\F_{3^{6}}$
  The base change of $A$ to $\F_{3^{6}}$ is 1.729.abb^3 $\times$ 1.729.cc. The endomorphism algebra for each factor is:

  • 1.729.abb^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• Endomorphism algebra over $\F_{3^{9}}$

  The base change of $A$ to $\F_{3^{9}}$ is 1.19683.a^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

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
4.3.a_d_j_a	$3$	(not in LMFDB)
4.3.d_d_j_bb	$3$	(not in LMFDB)
4.3.am_co_aii_rr	$9$	(not in LMFDB)