Abelian variety isogeny class 2.79.abg_pw over $\F_{79}$

• Label: 2.79.abg_pw
• Base field: $\F_{79}$
• 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:	$\F_{79}$
Dimension:	$2$
L-polynomial:	$1 - 32 x + 412 x^{2} - 2528 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.0643639783008$, $\pm0.193687002004$
Angle rank:	$2$ (numerical)
Number field:	4.0.84224.1
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4094$	$37722116$	$242693601422$	$1517142614746256$	$9468434513757436974$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$6042$	$492240$	$38950950$	$3077107888$	$243087971322$	$19203911465808$	$1517108801490174$	$119851595674284720$	$9468276079277166202$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=5x^6+18x^5+51x^4+36x^3+51x^2+10x+53$
• $y^2=34x^6+13x^5+63x^4+50x^3+9x^2+19x+48$
• $y^2=75x^6+36x^5+65x^4+57x^3+12x^2+71x+53$
• $y^2=48x^6+38x^5+5x^4+54x^3+18x^2+63x+65$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79}$.

Endomorphism algebra over $\F_{79}$

The endomorphism algebra of this simple isogeny class is 4.0.84224.1.

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
2.79.bg_pw	$2$	(not in LMFDB)