# Properties

 Label 2.4.ac_j Base Field $\F_{2^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $( 1 - x + 4 x^{2} )^{2}$ Frobenius angles: $\pm0.419569376745$, $\pm0.419569376745$ Angle rank: $1$ (numerical) Jacobians: 1

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

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

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2+(x^2+x)y=x^5+x^3+x^2+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 16 576 5776 57600 929296 16842816 276756496 4324377600 68311140496 1096007985216

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 31 87 223 903 4111 16887 65983 260583 1045231

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{2}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

 Subfield Primitive Model $\F_{2}$ 2.2.a_ab

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.a_h $2$ 2.16.o_dd 2.4.c_j $2$ 2.16.o_dd 2.4.b_ad $3$ 2.64.w_jp
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.a_h $2$ 2.16.o_dd 2.4.c_j $2$ 2.16.o_dd 2.4.b_ad $3$ 2.64.w_jp 2.4.a_ah $4$ 2.256.abi_bev 2.4.ab_ad $6$ (not in LMFDB)