Abelian variety isogeny class 2.81.c_dv over $\F_{3^{4}}$

• Label: 2.81.c_dv
• Base field: $\F_{3^{4}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x + 81 x^{2} )( 1 + 9 x + 81 x^{2} )$
	$1 + 2 x + 99 x^{2} + 162 x^{3} + 6561 x^{4}$
Frobenius angles:	$\pm0.372858997355$, $\pm0.666666666667$
Angle rank:	$1$ (numerical)
Jacobians:	$40$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6825$	$44342025$	$282375475200$	$1853317902297225$	$12157496427346895625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6756$	$531342$	$43053636$	$3486735924$	$282427692318$	$22876800478644$	$1853020317867396$	$150094634861442222$	$12157665457955226276$

Jacobians and polarizations

This isogeny class contains the Jacobians of 40 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{12}}$.

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

The isogeny class factors as 1.81.ah $\times$ 1.81.j and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.81.ah : \(\Q(\sqrt{-11}) \).
• 1.81.j : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{3^{4}}$

The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.cag. The endomorphism algebra for each factor is:

• 1.531441.acec : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.531441.cag : \(\Q(\sqrt{-11}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{4}}$.

Subfield	Primitive Model
$\F_{3}$	2.3.ae_j
$\F_{3}$	2.3.ac_d
$\F_{3}$	2.3.c_d
$\F_{3}$	2.3.e_j

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.81.aq_ir	$2$	(not in LMFDB)
2.81.ac_dv	$2$	(not in LMFDB)
2.81.q_ir	$2$	(not in LMFDB)
2.81.az_lc	$3$	(not in LMFDB)