Abelian variety isogeny class 2.4.ac_e over $\F_{2^{2}}$

• Label: 2.4.ac_e
• Base field: $\F_{2^{2}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 2 x + 4 x^{2} - 8 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.200000000000$, $\pm0.600000000000$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\zeta_{5})\)
Galois group:	$C_4$
Jacobians:	$2$

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11$	$341$	$3641$	$69905$	$1185921$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$21$	$57$	$273$	$1153$	$4161$	$16257$	$65793$	$261633$	$1044481$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2+y=(a+1)x^5+(a+1)x^3+a$
• $y^2+y=ax^5+ax^3+a+1$

Decomposition and endomorphism algebra

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

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

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\).

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

The base change of $A$ to $\F_{2^{10}}$ is 1.1024.cm^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is 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
2.4.c_e	$2$	2.16.e_q
2.4.i_y	$5$	2.1024.ey_jci
2.4.ai_y	$10$	(not in LMFDB)