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

• Label: 2.13.ae_be
• Base field: $\F_{13}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$( 1 - 2 x + 13 x^{2} )^{2}$
	$1 - 4 x + 30 x^{2} - 52 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.410543812489$, $\pm0.410543812489$
Angle rank:	$1$ (numerical)
Jacobians:	$8$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$144$	$36864$	$5143824$	$807469056$	$136968088464$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$214$	$2338$	$28270$	$368890$	$4825798$	$62777746$	$815802334$	$10604262634$	$137857087414$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=6 x^6+6 x^5+9 x^4+8 x^3+9 x^2+6 x+6$
• $y^2=6 x^6+3 x^5+5 x^4+8 x^3+11 x^2+x+7$
• $y^2=x^6+12$
• $y^2=4 x^6+x^4+x^2+4$
• $y^2=11 x^6+7 x^5+12 x^4+10 x^3+12 x^2+7 x+11$
• $y^2=7 x^6+7 x^4+7 x^2+7$
• $y^2=2 x^6+10 x^5+5 x^4+9 x^3+5 x^2+10 x+2$
• $y^2=x^6+3 x^3+1$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.a_w	$2$	2.169.bs_bfq
2.13.e_be	$2$	2.169.bs_bfq
2.13.ak_bz	$3$	(not in LMFDB)
2.13.ah_bk	$3$	(not in LMFDB)
2.13.c_aj	$3$	(not in LMFDB)
2.13.f_m	$3$	(not in LMFDB)
2.13.o_cx	$3$	(not in LMFDB)