# Properties

 Label 2.2.ac_c Base field $\F_{2}$ 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: $\F_{2}$ Dimension: $2$ L-polynomial: $1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4}$ Frobenius angles: $\pm0.0833333333333$, $\pm0.583333333333$ Angle rank: $0$ (numerical) Number field: $$\Q(\zeta_{12})$$ Galois group: $C_2^2$ Jacobians: 1

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is supersingular.

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

## Point counts

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2+y=x^5+x^3+1$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $13$ $25$ $169$ $1321$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $5$ $1$ $9$ $41$ $65$ $113$ $289$ $577$ $1025$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{12})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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 2.4.a_ae and its endomorphism algebra is $$\Q(\zeta_{12})$$.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.c_c$2$2.4.a_ae
2.2.e_i$3$2.8.ai_bg
2.2.ae_i$6$2.64.a_ey
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.c_c$2$2.4.a_ae
2.2.e_i$3$2.8.ai_bg
2.2.ae_i$6$2.64.a_ey
2.2.a_a$6$2.64.a_ey
2.2.a_ac$8$2.256.bg_bdo
2.2.a_c$8$2.256.bg_bdo
2.2.ac_e$24$(not in LMFDB)
2.2.a_ae$24$(not in LMFDB)
2.2.a_e$24$(not in LMFDB)
2.2.c_e$24$(not in LMFDB)