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

• Label: 2.13.a_w
• 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:	$( 1 - 2 x + 13 x^{2} )( 1 + 2 x + 13 x^{2} )$
	$1 + 22 x^{2} + 169 x^{4}$
Frobenius angles:	$\pm0.410543812489$, $\pm0.589456187511$
Angle rank:	$1$ (numerical)
Jacobians:	$22$
Isomorphism classes:	128

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$192$	$36864$	$4826304$	$807469056$	$137857789632$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$214$	$2198$	$28270$	$371294$	$4825798$	$62748518$	$815802334$	$10604499374$	$137857087414$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 22 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{2}}$.

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ac $\times$ 1.13.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.13.ac : \(\Q(\sqrt{-3}) \).
• 1.13.c : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{13^{2}}$ is 1.169.w^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.ae_be	$2$	2.169.bs_bfq
2.13.e_be	$2$	2.169.bs_bfq
2.13.am_cj	$3$	(not in LMFDB)
2.13.aj_bo	$3$	(not in LMFDB)
2.13.ad_q	$3$	(not in LMFDB)
2.13.a_ax	$3$	(not in LMFDB)
2.13.a_b	$3$	(not in LMFDB)
2.13.d_q	$3$	(not in LMFDB)
2.13.j_bo	$3$	(not in LMFDB)
2.13.m_cj	$3$	(not in LMFDB)