# Properties

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

# Related objects

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 + x^{2} + 4 x^{4}$ Frobenius angles: $\pm0.290215311628$, $\pm0.709784688372$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{3}, \sqrt{-5})$$ Galois group: $C_2^2$ Jacobians: 1

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

## Newton polygon

This isogeny class is ordinary. $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

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

• $y^2+(x^2+x+1)y=x^5+x^3+x^2+1$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $36$ $54$ $576$ $1086$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $7$ $9$ $31$ $33$ $43$ $129$ $223$ $513$ $1147$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{3}, \sqrt{-5})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{2}}$ is 1.4.b 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.a_ab$4$2.16.o_dd
2.2.ad_f$12$(not in LMFDB)
2.2.d_f$12$(not in LMFDB)