Abelian variety isogeny class 3.13.ap_eg_atb over $\F_{13}$

• Label: 3.13.ap_eg_atb
• Base field: $\F_{13}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{13}$
Dimension:	$3$
L-polynomial:	$( 1 - 7 x + 13 x^{2} )( 1 - 5 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$
	$1 - 15 x + 110 x^{2} - 495 x^{3} + 1430 x^{4} - 2535 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.0772104791556$, $\pm0.256122854178$, $\pm0.363422825076$
Angle rank:	$2$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$693$	$4700619$	$11042583552$	$23457973758219$	$51092996893356033$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$165$	$2288$	$28757$	$370619$	$4822092$	$62740103$	$815749637$	$10604635184$	$137858890125$

Jacobians and polarizations

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

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.af $\times$ 1.13.ad 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.af : \(\Q(\sqrt{-3}) \).
• 1.13.ad : \(\Q(\sqrt{-43}) \).

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

The base change of $A$ to $\F_{13^{6}}$ is 1.4826809.afmo $\times$ 1.4826809.atm^2. The endomorphism algebra for each factor is:

• 1.4826809.afmo : \(\Q(\sqrt{-43}) \).
• 1.4826809.atm^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.b $\times$ 1.169.r. The endomorphism algebra for each factor is:

  • 1.169.ax : \(\Q(\sqrt{-3}) \).
  • 1.169.b : \(\Q(\sqrt{-3}) \).
  • 1.169.r : \(\Q(\sqrt{-43}) \).
• Endomorphism algebra over $\F_{13^{3}}$
  The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs $\times$ 1.2197.cs $\times$ 1.2197.dm. The endomorphism algebra for each factor is:

  • 1.2197.acs : \(\Q(\sqrt{-3}) \).
  • 1.2197.cs : \(\Q(\sqrt{-3}) \).
  • 1.2197.dm : \(\Q(\sqrt{-43}) \).

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
3.13.aj_bm_aez	$2$	(not in LMFDB)
3.13.af_k_az	$2$	(not in LMFDB)
3.13.ab_ac_db	$2$	(not in LMFDB)
3.13.b_ac_adb	$2$	(not in LMFDB)
3.13.f_k_z	$2$	(not in LMFDB)
3.13.j_bm_ez	$2$	(not in LMFDB)
3.13.p_eg_tb	$2$	(not in LMFDB)
3.13.am_dc_anq	$3$	(not in LMFDB)
3.13.ag_bm_aew	$3$	(not in LMFDB)
3.13.ad_ak_cr	$3$	(not in LMFDB)
3.13.ad_o_ad	$3$	(not in LMFDB)
3.13.ad_bj_aco	$3$	(not in LMFDB)
3.13.a_u_be	$3$	(not in LMFDB)
3.13.g_ba_ek	$3$	(not in LMFDB)