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

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

Invariants

Base field:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$1 + 169 x^{4}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.750000000000$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(i, \sqrt{26})\)
Galois group:	$C_2^2$
Jacobians:	$21$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$170$	$28900$	$4826810$	$835210000$	$137858491850$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$170$	$2198$	$29238$	$371294$	$4826810$	$62748518$	$815616478$	$10604499374$	$137858491850$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):

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

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 \(\Q(i, \sqrt{26})\).

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

The base change of $A$ to $\F_{13^{4}}$ is 1.28561.na^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.a_na and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{-1}) \) with the following ramification data at primes above $13$, and unramified at all archimedean places:

  $v$	($ 13 $,\( \pi + 5 \))	($ 13 $,\( \pi + 8 \))
  $\operatorname{inv}_v$	$1/2$	$1/2$

  where $\pi$ is a root of $x^{2} + 1$.

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_aba	$8$	(not in LMFDB)
2.13.a_ba	$8$	(not in LMFDB)
2.13.a_an	$24$	(not in LMFDB)