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

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

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$1 - 5 x - 4 x^{2} - 145 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0129947766296$, $\pm0.679661443296$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-91})\)
Galois group:	$C_2^2$
Jacobians:	$2$
Isomorphism classes:	12
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$688$	$679744$	$579846400$	$499826639104$	$420583669410928$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$809$	$23770$	$706689$	$20505125$	$594728678$	$17249814425$	$500245350049$	$14507131757410$	$420707228578529$

Jacobians and polarizations

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

• $y^2=25 x^6+26 x^5+2 x^4+13 x^3+15 x^2+28 x+22$
• $y^2=28 x^6+27 x^5+10 x^4+24 x^3+8 x^2+22 x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29^{3}}$.

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-91})\).

Endomorphism algebra over $\overline{\F}_{29}$

The base change of $A$ to $\F_{29^{3}}$ is 1.24389.aly^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$

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.f_ae	$2$	(not in LMFDB)
2.29.k_df	$3$	(not in LMFDB)
2.29.ak_df	$6$	(not in LMFDB)