Abelian variety isogeny class 6.2.ag_r_abd_bg_ay_u over $\F_{2}$

• Label: 6.2.ag_r_abd_bg_ay_u
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $3$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )( 1 - 2 x^{2} + 4 x^{4} )( 1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6} )$
	$1 - 6 x + 17 x^{2} - 29 x^{3} + 32 x^{4} - 24 x^{5} + 20 x^{6} - 48 x^{7} + 128 x^{8} - 232 x^{9} + 272 x^{10} - 192 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.0435981566527$, $\pm0.166666666667$, $\pm0.250000000000$, $\pm0.329312442367$, $\pm0.527830414776$, $\pm0.833333333333$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3$	$3195$	$443313$	$22700475$	$1043107773$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$3$	$12$	$23$	$32$	$84$	$74$	$231$	$543$	$1028$

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 $\F_{2^{168}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac $\times$ 2.2.a_ac $\times$ 3.2.ae_j_ap and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.ac : \(\Q(\sqrt{-1}) \).
• 2.2.a_ac : \(\Q(\sqrt{-2}, \sqrt{-3})\).
• 3.2.ae_j_ap : \(\Q(\zeta_{7})\).

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{168}}$ is 1.374144419156711147060143317175368453031918731001856.abidcukshtnepxypydvc^3 $\times$ 1.374144419156711147060143317175368453031918731001856.rwsuotmkbtwngpmnyl^3. The endomorphism algebra for each factor is:

• 1.374144419156711147060143317175368453031918731001856.abidcukshtnepxypydvc^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.374144419156711147060143317175368453031918731001856.rwsuotmkbtwngpmnyl^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.ac^2 $\times$ 1.4.a $\times$ 3.4.c_ad_an. The endomorphism algebra for each factor is:

  • 1.4.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 3.4.c_ad_an : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.a^2 $\times$ 1.8.e $\times$ 3.8.ab_ag_bb. The endomorphism algebra for each factor is:

  • 1.8.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
  • 1.8.e : \(\Q(\sqrt{-1}) \).
  • 3.8.ab_ag_bb : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.e^2 $\times$ 1.16.i $\times$ 3.16.ak_bl_adt. The endomorphism algebra for each factor is:

  • 1.16.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 3.16.ak_bl_adt : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.q^2 $\times$ 3.64.an_ag_bcp. The endomorphism algebra for each factor is:

  • 1.64.a : \(\Q(\sqrt{-1}) \).
  • 1.64.q^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 3.64.an_ag_bcp : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{7}}$
  The base change of $A$ to $\F_{2^{7}}$ is 1.128.aq $\times$ 1.128.an^3 $\times$ 2.128.a_aey. The endomorphism algebra for each factor is:

  • 1.128.aq : \(\Q(\sqrt{-1}) \).
  • 1.128.an^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 2.128.a_aey : \(\Q(\sqrt{-2}, \sqrt{-3})\).
• Endomorphism algebra over $\F_{2^{8}}$
  The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg $\times$ 1.256.q^2 $\times$ 3.256.aba_xp_amtd. The endomorphism algebra for each factor is:

  • 1.256.abg : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.256.q^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.256.aba_xp_amtd : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{12}}$
  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey^2 $\times$ 1.4096.ey $\times$ 3.4096.agz_bbmw_adccgn. The endomorphism algebra for each factor is:

  • 1.4096.aey^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 3.4096.agz_bbmw_adccgn : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{14}}$
  The base change of $A$ to $\F_{2^{14}}$ is 1.16384.aey^2 $\times$ 1.16384.a $\times$ 1.16384.dj^3. The endomorphism algebra for each factor is:

  • 1.16384.aey^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.16384.a : \(\Q(\sqrt{-1}) \).
  • 1.16384.dj^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
• Endomorphism algebra over $\F_{2^{21}}$
  The base change of $A$ to $\F_{2^{21}}$ is 1.2097152.a^2 $\times$ 1.2097152.dau $\times$ 1.2097152.edn^3. The endomorphism algebra for each factor is:

  • 1.2097152.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
  • 1.2097152.dau : \(\Q(\sqrt{-1}) \).
  • 1.2097152.edn^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
• Endomorphism algebra over $\F_{2^{24}}$
  The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc^3 $\times$ 3.16777216.gnr_azynys_aocxfgarh. The endomorphism algebra for each factor is:

  • 1.16777216.amdc^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 3.16777216.gnr_azynys_aocxfgarh : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{28}}$
  The base change of $A$ to $\F_{2^{28}}$ is 1.268435456.yge^2 $\times$ 1.268435456.blhf^3 $\times$ 1.268435456.bwmi. The endomorphism algebra for each factor is:

  • 1.268435456.yge^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.268435456.blhf^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 1.268435456.bwmi : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{42}}$
  The base change of $A$ to $\F_{2^{42}}$ is 1.4398046511104.ahxvrd^3 $\times$ 1.4398046511104.a $\times$ 1.4398046511104.jeqpk^2. The endomorphism algebra for each factor is:

  • 1.4398046511104.ahxvrd^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 1.4398046511104.a : \(\Q(\sqrt{-1}) \).
  • 1.4398046511104.jeqpk^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{56}}$
  The base change of $A$ to $\F_{2^{56}}$ is 1.72057594037927936.abtevrtg $\times$ 1.72057594037927936.aigsnzt^3 $\times$ 1.72057594037927936.wpkvwq^2. The endomorphism algebra for each factor is:

  • 1.72057594037927936.abtevrtg : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.72057594037927936.aigsnzt^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 1.72057594037927936.wpkvwq^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• Endomorphism algebra over $\F_{2^{84}}$
  The base change of $A$ to $\F_{2^{84}}$ is 1.19342813113834066795298816.abqdecdesiy^2 $\times$ 1.19342813113834066795298816.auojdvfkpl^3 $\times$ 1.19342813113834066795298816.bqdecdesiy. The endomorphism algebra for each factor is:

  • 1.19342813113834066795298816.abqdecdesiy^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.19342813113834066795298816.auojdvfkpl^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 1.19342813113834066795298816.bqdecdesiy : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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
6.2.ac_b_ab_e_ai_m	$2$	(not in LMFDB)
6.2.c_b_b_e_i_m	$2$	(not in LMFDB)
6.2.g_r_bd_bg_y_u	$2$	(not in LMFDB)
6.2.ag_x_acn_fq_akk_qa	$3$	(not in LMFDB)

Additional information

This isogeny class is an example of one having the largest degree endomorphism field in the database (168).