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

• Label: 2.13.ag_s
• 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:	$1 - 6 x + 18 x^{2} - 78 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.0497783698316$, $\pm0.549778369832$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{17})\)
Galois group:	$C_2^2$
Jacobians:	$8$
Isomorphism classes:	14

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$104$	$28288$	$4557800$	$800210944$	$137856264104$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$170$	$2072$	$28014$	$371288$	$4826810$	$62727176$	$815694814$	$10604669096$	$137858491850$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{4}}$.

Endomorphism algebra over $\F_{13}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{17})\).

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

The base change of $A$ to $\F_{13^{4}}$ is 1.28561.ako^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-17}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{13^{2}}$

  The base change of $A$ to $\F_{13^{2}}$ is the simple isogeny class 2.169.a_ako and its endomorphism algebra is \(\Q(i, \sqrt{17})\).

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.g_s	$2$	2.169.a_ako
2.13.a_ai	$8$	(not in LMFDB)
2.13.a_i	$8$	(not in LMFDB)