Abelian variety isogeny class 4.4.ak_bw_afw_no over $\F_{2^{2}}$

• Label: 4.4.ak_bw_afw_no
• Base field: $\F_{2^{2}}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x )^{4}( 1 - 2 x + 4 x^{2} )( 1 + 4 x^{2} )$
	$1 - 10 x + 48 x^{2} - 152 x^{3} + 352 x^{4} - 608 x^{5} + 768 x^{6} - 640 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $0$, $0$, $\pm0.333333333333$, $\pm0.5$
Angle rank:	$0$ (numerical)

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$15$	$42525$	$12641265$	$3109640625$	$939982761825$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$13$	$49$	$177$	$865$	$3841$	$15745$	$64257$	$261121$	$1047553$

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 $\F_{2^{24}}$.

Endomorphism algebra over $\F_{2^{2}}$

The isogeny class factors as 1.4.ae^2 $\times$ 1.4.ac $\times$ 1.4.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.4.ae^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4.ac : \(\Q(\sqrt{-3}) \).
• 1.4.a : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc^4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai^2 $\times$ 1.16.e $\times$ 1.16.i. The endomorphism algebra for each factor is:

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

  • 1.64.aq^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.64.a : \(\Q(\sqrt{-1}) \).
  • 1.64.q : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{8}}$
  The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^3 $\times$ 1.256.q. The endomorphism algebra for each factor is:

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

  • 1.4096.aey^3 : $\mathrm{M}_{3}(B)$, 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 $\infty$.

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
4.4.ag_q_abo_ds	$2$	(not in LMFDB)
4.4.ac_a_i_abg	$2$	(not in LMFDB)
4.4.c_a_ai_abg	$2$	(not in LMFDB)
4.4.g_q_bo_ds	$2$	(not in LMFDB)
4.4.k_bw_fw_no	$2$	(not in LMFDB)
4.4.ae_a_q_abg	$3$	(not in LMFDB)
4.4.ae_m_abg_cm	$3$	(not in LMFDB)
4.4.c_m_q_cm	$3$	(not in LMFDB)
4.4.i_bk_ei_jw	$3$	(not in LMFDB)