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

• Label: 2.49.av_hs
• 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:	$1 - 21 x + 200 x^{2} - 1029 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.0956797089624$, $\pm0.316591506724$
Angle rank:	$2$ (numerical)
Number field:	4.0.40293.1
Galois group:	$D_{4}$
Jacobians:	$16$
Isomorphism classes:	16

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1552$	$5667904$	$13870882048$	$33241508796672$	$79789176320595472$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$29$	$2361$	$117902$	$5766289$	$282464309$	$13841117790$	$678222628469$	$33232940291425$	$1628413742889326$	$79792267347293961$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.40293.1.

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.v_hs	$2$	(not in LMFDB)