Abelian variety isogeny class 2.13.ad_ae over $\F_{13}$

• Label: 2.13.ad_ae
• Base field: $\F_{13}$
• 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:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$1 - 3 x - 4 x^{2} - 39 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.0300894917430$, $\pm0.696756158410$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-43})\)
Galois group:	$C_2^2$
Jacobians:	$1$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$124$	$25792$	$4443664$	$814305024$	$137479333324$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$153$	$2018$	$28513$	$370271$	$4819398$	$62750699$	$815676001$	$10604227754$	$137858795793$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=6 x^6+4 x^5+10 x^4+7 x^3+8 x^2+10 x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{3}}$.

Endomorphism algebra over $\F_{13}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-43})\).

Endomorphism algebra over $\overline{\F}_{13}$

The base change of $A$ to $\F_{13^{3}}$ is 1.2197.adm^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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
2.13.d_ae	$2$	2.169.ar_eq
2.13.g_bj	$3$	(not in LMFDB)
2.13.ag_bj	$6$	(not in LMFDB)