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

• Label: 2.13.aj_bo
• 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:	$( 1 - 7 x + 13 x^{2} )( 1 - 2 x + 13 x^{2} )$
	$1 - 9 x + 40 x^{2} - 117 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.0772104791556$, $\pm0.410543812489$
Angle rank:	$1$ (numerical)
Jacobians:	$4$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$84$	$28224$	$4826304$	$806190336$	$137254909764$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$169$	$2198$	$28225$	$369665$	$4825798$	$62765141$	$815787169$	$10604499374$	$137858349889$

Jacobians and polarizations

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

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

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.ah $\times$ 1.13.ac 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.ah : \(\Q(\sqrt{-3}) \).
• 1.13.ac : \(\Q(\sqrt{-3}) \).

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.ax $\times$ 1.169.w. The endomorphism algebra for each factor is:

  • 1.169.ax : \(\Q(\sqrt{-3}) \).
  • 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.af_m	$2$	2.169.ab_agm
2.13.f_m	$2$	2.169.ab_agm
2.13.am_cj	$3$	(not in LMFDB)
2.13.ad_q	$3$	(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)