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

• Label: 2.29.ar_ew
• 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:	$1 - 17 x + 126 x^{2} - 493 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0627763123564$, $\pm0.296043746988$
Angle rank:	$2$ (numerical)
Number field:	4.0.95948.1
Galois group:	$D_{4}$
Jacobians:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$458$	$676924$	$595625336$	$500403882368$	$420616263537538$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$805$	$24424$	$707505$	$20506713$	$594767674$	$17249570701$	$500245854273$	$14507151281944$	$420707290033005$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=20 x^6+17 x^5+23 x^4+26 x^3+x^2+9 x+6$
• $y^2=12 x^6+20 x^5+26 x^4+9 x^3+18 x^2+2 x+19$

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 4.0.95948.1.

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.r_ew	$2$	(not in LMFDB)