Abelian variety isogeny class 2.9.ae_i over $\F_{3^{2}}$

• Label: 2.9.ae_i
• Base field: $\F_{3^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 4 x + 8 x^{2} - 36 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.0937471905441$, $\pm0.593747190544$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{14})\)
Galois group:	$C_2^2$
Jacobians:	$6$
Isomorphism classes:	8

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$50$	$6500$	$478850$	$42250000$	$3512625250$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$82$	$654$	$6438$	$59486$	$531442$	$4781174$	$43065278$	$387473766$	$3486784402$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=(a+2)x^6+(a+2)x^5+(a+1)x^4+(a+1)x^2+(2a+1)x+a+2$
• $y^2=(2a+1)x^6+(a+2)x^5+(a+1)x^4+(a+1)x^2+(2a+1)x+2a+1$
• $y^2=(a+2)x^6+(a+2)x^5+ax^4+(2a+1)x^3+x$
• $y^2=(2a+2)x^6+(a+1)x^5+(2a+1)x^4+(a+2)x^3+x^2+(2a+2)x+2a+2$
• $y^2=(a+2)x^5+(a+1)x^4+ax^3+(a+2)x^2+(a+2)x+a+2$
• $y^2=2ax^6+2ax^5+(2a+1)x^4+ax^3+x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{8}}$.

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

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{14})\).

Endomorphism algebra over $\overline{\F}_{3^{2}}$

The base change of $A$ to $\F_{3^{8}}$ is 1.6561.ack^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-14}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{4}}$

  The base change of $A$ to $\F_{3^{4}}$ is the simple isogeny class 2.81.a_ack and its endomorphism algebra is \(\Q(i, \sqrt{14})\).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.9.e_i	$2$	2.81.a_ack
2.9.a_ak	$8$	(not in LMFDB)
2.9.a_k	$8$	(not in LMFDB)