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

• Label: 2.29.ap_eb
• 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:	$1 - 15 x + 105 x^{2} - 435 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0657493903052$, $\pm0.364141013558$
Angle rank:	$2$ (numerical)
Number field:	4.0.642061.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$497$	$694309$	$595893557$	$499603232821$	$420455022724352$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$827$	$24435$	$706371$	$20498850$	$594766667$	$17249883855$	$500247890563$	$14507153047035$	$420707235148502$

Jacobians and polarizations

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

• $y^2=27 x^6+28 x^5+8 x^4+28 x^3+18 x^2+18 x+8$
• $y^2=10 x^6+17 x^5+22 x^4+2 x^3+2 x^2+5 x+8$
• $y^2=11 x^6+4 x^5+24 x^4+19 x^3+15 x^2+7 x+10$
• $y^2=20 x^6+4 x^5+12 x^3+5 x^2+2$

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 4.0.642061.1.

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.p_eb	$2$	(not in LMFDB)