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

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

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x )^{2}( 1 + 4 x^{2} )( 1 - 2 x + 3 x^{2} - 8 x^{3} + 16 x^{4} )$
	$1 - 6 x + 19 x^{2} - 52 x^{3} + 120 x^{4} - 208 x^{5} + 304 x^{6} - 384 x^{7} + 256 x^{8}$
Frobenius angles:	$0$, $0$, $\pm0.168977707736$, $\pm0.5$, $\pm0.618033150523$
Angle rank:	$2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$50$	$67500$	$10542350$	$3523500000$	$1157453626250$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$19$	$35$	$207$	$1079$	$4147$	$16211$	$65247$	$260855$	$1045699$

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 $\F_{2^{8}}$.

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

The isogeny class factors as 1.4.ae $\times$ 1.4.a $\times$ 2.4.ac_d 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 : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.4.a : \(\Q(\sqrt{-1}) \).
• 2.4.ac_d : 4.0.32832.1.

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^2 $\times$ 2.256.o_rx. The endomorphism algebra for each factor is:

• 1.256.abg^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.256.o_rx : 4.0.32832.1.

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 $\times$ 1.16.i $\times$ 2.16.c_j. The endomorphism algebra for each factor is:

  • 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.16.c_j : 4.0.32832.1.

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.ac_d_ae_ai	$2$	(not in LMFDB)
4.4.c_d_e_ai	$2$	(not in LMFDB)
4.4.g_t_ca_eq	$2$	(not in LMFDB)
4.4.a_h_ak_y	$3$	(not in LMFDB)