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

• Label: 2.13.a_g
• 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:	$1 + 6 x^{2} + 169 x^{4}$
Frobenius angles:	$\pm0.287062121659$, $\pm0.712937878341$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{5})\)
Galois group:	$C_2^2$
Jacobians:	$32$
Isomorphism classes:	72

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$176$	$30976$	$4823984$	$833130496$	$137859173936$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$182$	$2198$	$29166$	$371294$	$4821158$	$62748518$	$815662558$	$10604499374$	$137859856022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{13}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{5})\).

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

The base change of $A$ to $\F_{13^{2}}$ is 1.169.g^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.13.a_ag	$4$	(not in LMFDB)
2.13.ai_bg	$8$	(not in LMFDB)
2.13.i_bg	$8$	(not in LMFDB)