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

• Label: 2.13.d_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:	$\pm0.303243841590$, $\pm0.969910508257$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-43})\)
Galois group:	$C_2^2$
Jacobians:	$1$
Isomorphism classes:	3
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$208$	$25792$	$5234944$	$814305024$	$138239000848$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$17$	$153$	$2378$	$28513$	$372317$	$4819398$	$62746337$	$815676001$	$10604770994$	$137858795793$

Jacobians and polarizations

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

• $y^2=12 x^6+8 x^5+7 x^4+x^3+3 x^2+7 x$

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.dm^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.ad_ae	$2$	2.169.ar_eq
2.13.ag_bj	$3$	(not in LMFDB)
2.13.a_r	$6$	(not in LMFDB)