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

• Label: 2.13.ae_ba
• Base field: $\F_{13}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 - 4 x + 13 x^{2} )( 1 + 13 x^{2} )$
	$1 - 4 x + 26 x^{2} - 52 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.312832958189$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$20$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$140$	$35280$	$5033420$	$812851200$	$137768638700$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$206$	$2290$	$28462$	$371050$	$4827134$	$62735410$	$815674078$	$10604671690$	$137859917486$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

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

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.ae $\times$ 1.13.a 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.ae : \(\Q(\sqrt{-1}) \).
• 1.13.a : \(\Q(\sqrt{-13}) \).

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

The base change of $A$ to $\F_{13^{2}}$ is 1.169.k $\times$ 1.169.ba. The endomorphism algebra for each factor is:

• 1.169.k : \(\Q(\sqrt{-1}) \).
• 1.169.ba : the quaternion algebra over \(\Q\) ramified at $13$ and $\infty$.

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.e_ba	$2$	2.169.bk_xa
2.13.ag_ba	$4$	(not in LMFDB)
2.13.g_ba	$4$	(not in LMFDB)