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

• Label: 2.13.ag_x
• 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:	$1 - 6 x + 23 x^{2} - 78 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.146166291522$, $\pm0.520500375144$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{12})\)
Galois group:	$C_2^2$
Jacobians:	$12$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$109$	$30193$	$4752400$	$808961049$	$138302871229$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$180$	$2162$	$28324$	$372488$	$4834950$	$62757416$	$815730244$	$10604724986$	$137859174900$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

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

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

The base change of $A$ to $\F_{13^{3}}$ is 1.2197.as^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
2.13.g_x	$2$	2.169.k_acr
2.13.m_ck	$3$	(not in LMFDB)
2.13.ae_d	$4$	(not in LMFDB)