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

• Label: 2.13.a_ba
• Base field: $\F_{13}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$( 1 + 13 x^{2} )^{2}$
	$1 + 26 x^{2} + 169 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$4$
Cyclic group of points:	no
Non-cyclic primes:	$2, 7$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$196$	$38416$	$4831204$	$796594176$	$137859234436$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$222$	$2198$	$27886$	$371294$	$4835598$	$62748518$	$815616478$	$10604499374$	$137859977022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

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

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.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$

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

The base change of $A$ to $\F_{13^{2}}$ is 1.169.ba^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $13$ and $\infty$.

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.a_an	$3$	(not in LMFDB)
2.13.a_aba	$4$	(not in LMFDB)
2.13.a_a	$8$	(not in LMFDB)