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

• Label: 2.29.as_fi
• Base field: $\F_{29}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 29 x^{2} )( 1 - 8 x + 29 x^{2} )$
	$1 - 18 x + 138 x^{2} - 522 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.121118941591$, $\pm0.233506187634$
Angle rank:	$2$ (numerical)
Jacobians:	$4$
Isomorphism classes:	14

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$440$	$668800$	$596165240$	$501353881600$	$420928448646200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$794$	$24444$	$708846$	$20521932$	$594870122$	$17250001788$	$500246526046$	$14507146057356$	$420707245302074$

Jacobians and polarizations

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

• $y^2=2x^6+2x^4+21x^3+15x^2+10$
• $y^2=17x^6+22x^5+5x^4+x^3+5x^2+22x+17$
• $y^2=3x^6+25x^5+14x^4+16x^3+14x^2+25x+3$
• $y^2=3x^6+11x^4+13x^3+19x^2+14$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ak $\times$ 1.29.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.29.ak : \(\Q(\sqrt{-1}) \).
• 1.29.ai : \(\Q(\sqrt{-13}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.29.ac_aw	$2$	(not in LMFDB)
2.29.c_aw	$2$	(not in LMFDB)
2.29.s_fi	$2$	(not in LMFDB)