Abelian variety isogeny class 2.49.au_hb over $\F_{7^{2}}$

• Label: 2.49.au_hb
• Base field: $\F_{7^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 20 x + 183 x^{2} - 980 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.0429101801095$, $\pm0.355813008897$
Angle rank:	$2$ (numerical)
Number field:	4.0.2019600.2
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	24

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1585$	$5682225$	$13845805540$	$33215792853225$	$79777550580594625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$2368$	$117690$	$5761828$	$282423150$	$13840910278$	$678222058110$	$33232935334468$	$1628413634611770$	$79792266070217248$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7^{2}}$

The endomorphism algebra of this simple isogeny class is 4.0.2019600.2.

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
2.49.u_hb	$2$	(not in LMFDB)