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

• Label: 2.13.ad_q
• Base field: $\F_{13}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• 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:	$( 1 - 5 x + 13 x^{2} )( 1 + 2 x + 13 x^{2} )$
	$1 - 3 x + 16 x^{2} - 39 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.256122854178$, $\pm0.589456187511$
Angle rank:	$1$ (numerical)
Jacobians:	$14$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$144$	$32832$	$4826304$	$821193984$	$138594214224$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$193$	$2198$	$28753$	$373271$	$4825798$	$62721299$	$815710081$	$10604499374$	$137857931593$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.af $\times$ 1.13.c 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.af : \(\Q(\sqrt{-3}) \).
• 1.13.c : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{13^{6}}$ is 1.4826809.atm^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{13^{2}}$
  The base change of $A$ to $\F_{13^{2}}$ is 1.169.b $\times$ 1.169.w. The endomorphism algebra for each factor is:

  • 1.169.b : \(\Q(\sqrt{-3}) \).
  • 1.169.w : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{13^{3}}$
  The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs $\times$ 1.2197.cs. The endomorphism algebra for each factor is:

  • 1.2197.acs : \(\Q(\sqrt{-3}) \).
  • 1.2197.cs : \(\Q(\sqrt{-3}) \).

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.ah_bk	$2$	2.169.x_nw
2.13.h_bk	$2$	2.169.x_nw
2.13.am_cj	$3$	(not in LMFDB)
2.13.aj_bo	$3$	(not in LMFDB)
2.13.a_ax	$3$	(not in LMFDB)
2.13.a_b	$3$	(not in LMFDB)
2.13.a_w	$3$	(not in LMFDB)
2.13.d_q	$3$	(not in LMFDB)
2.13.j_bo	$3$	(not in LMFDB)
2.13.m_cj	$3$	(not in LMFDB)