Abelian variety isogeny class 2.29.an_df over $\F_{29}$

• Label: 2.29.an_df
• Base field: $\F_{29}$
• 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:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$1 - 13 x + 83 x^{2} - 377 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0469645722643$, $\pm0.430084384264$
Angle rank:	$2$ (numerical)
Number field:	4.0.1318797.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$535$	$703525$	$592598635$	$498501633925$	$420404419718800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$17$	$839$	$24299$	$704811$	$20496382$	$594804647$	$17250004243$	$500246136691$	$14507137145321$	$420707205369974$

Jacobians and polarizations

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

• $y^2=27 x^6+14 x^5+10 x^4+8 x^3+5 x^2+11 x+12$
• $y^2=17 x^6+11 x^5+17 x^4+12 x^3+24 x^2+19 x+11$
• $y^2=18 x^6+14 x^5+21 x^4+5 x^3+18 x^2+8 x+6$
• $y^2=28 x^6+9 x^5+13 x^3+12 x^2+18 x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29}$.

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is 4.0.1318797.1.

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
2.29.n_df	$2$	(not in LMFDB)