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

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

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$1 + 4 x - 13 x^{2} + 116 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.287785608258$, $\pm0.954452274924$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{12})\)
Galois group:	$C_2^2$
Jacobians:	$0$

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$949$	$672841$	$608806276$	$500303709529$	$420882943619749$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$800$	$24958$	$707364$	$20519714$	$594759566$	$17249755466$	$500245005124$	$14507150229622$	$420707265620000$

Jacobians and polarizations

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

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 \(\Q(\zeta_{12})\).

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

The base change of $A$ to $\F_{29^{3}}$ is 1.24389.ky^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.ae_an	$2$	(not in LMFDB)
2.29.ai_cw	$3$	(not in LMFDB)
2.29.ak_ct	$4$	(not in LMFDB)