Abelian variety isogeny class 4.2.af_n_az_bn over $\F_{2}$

• Label: 4.2.af_n_az_bn
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: no
• Contains a Jacobian: no

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Invariants

Base field:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 50 x^{5} + 52 x^{6} - 40 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.00978468837242$, $\pm0.190215311628$, $\pm0.409784688372$, $\pm0.609784688372$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{15})\)
Galois group:	$C_4\times C_2$
Jacobians:	$0$
Isomorphism classes:	1
Cyclic group of points:	yes

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$241$	$1891$	$43621$	$929296$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$6$	$4$	$10$	$33$	$54$	$124$	$274$	$418$	$781$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{10}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{15})\).

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

The base change of $A$ to $\F_{2^{10}}$ is 1.1024.acj^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 4.4.b_ad_ah_f and its endomorphism algebra is \(\Q(\zeta_{15})\).

• Endomorphism algebra over $\F_{2^{5}}$

  The base change of $A$ to $\F_{2^{5}}$ is 2.32.a_acj^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$

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
4.2.f_n_z_bn	$2$	4.4.b_ad_ah_f
4.2.b_e_c_j	$3$	(not in LMFDB)
4.2.e_e_ah_av	$3$	(not in LMFDB)

Additional information

This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.