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

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

Invariants

Base field:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$1 - 13 x^{2} + 169 x^{4}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.833333333333$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-13})\)
Galois group:	$C_2^2$
Jacobians:	$0$

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular.

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$157$	$24649$	$4831204$	$825470361$	$137858120557$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$144$	$2198$	$28900$	$371294$	$4835598$	$62748518$	$815787844$	$10604499374$	$137857749264$

Jacobians and polarizations

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{6}}$.

Endomorphism algebra over $\F_{13}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-13})\).

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

The base change of $A$ to $\F_{13^{6}}$ is 1.4826809.gna^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $13$ and $\infty$.

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.aba_tn and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{-3}) \) with the following ramification data at primes above $13$, and unramified at all archimedean places:

  $v$	($ 13 $,\( \pi + 3 \))	($ 13 $,\( \pi + 9 \))
  $\operatorname{inv}_v$	$1/2$	$1/2$

  where $\pi$ is a root of $x^{2} - x + 1$.
• Endomorphism algebra over $\F_{13^{3}}$

  The base change of $A$ to $\F_{13^{3}}$ is 1.2197.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$

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_ba	$3$	(not in LMFDB)
2.13.a_n	$4$	(not in LMFDB)
2.13.a_aba	$12$	(not in LMFDB)