# Properties

 Label 2.2.ac_e Base field $\F_{2}$ Dimension $2$ $p$-rank $0$ Ordinary no Supersingular yes Simple no Geometrically simple no Primitive yes Principally polarizable yes Contains a Jacobian yes

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 + 2 x^{2} )$ $1 - 2x + 4x^{2} - 4x^{3} + 4x^{4}$ Frobenius angles: $\pm0.250000000000$, $\pm0.5$ Angle rank: $0$ (numerical) Jacobians: 1

This isogeny class is not 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:

• $y^2+y=x^5+x^4+1$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3$ $45$ $117$ $225$ $1353$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $9$ $13$ $17$ $41$ $81$ $113$ $193$ $481$ $1089$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 1.2.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 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 $\F_{2^{8}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 1.4.e. The endomorphism algebra for each factor is: 1.4.a : $$\Q(\sqrt{-1})$$. 1.4.e : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i. 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 $\infty$.

## 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_e$2$2.4.e_i
2.2.ae_i$8$2.256.acm_chc
2.2.a_ae$8$2.256.acm_chc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.c_e$2$2.4.e_i
2.2.ae_i$8$2.256.acm_chc
2.2.a_ae$8$2.256.acm_chc
2.2.a_a$8$2.256.acm_chc
2.2.a_e$8$2.256.acm_chc
2.2.e_i$8$2.256.acm_chc
2.2.ac_c$24$(not in LMFDB)
2.2.a_ac$24$(not in LMFDB)
2.2.a_c$24$(not in LMFDB)
2.2.c_c$24$(not in LMFDB)