# Properties

 Label 2.4.ad_f 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: $1 - 3 x + 5 x^{2} - 12 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.103279877171$, $\pm0.563386789496$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Galois group: $C_2^2$ Jacobians: 2

This isogeny class is simple but not geometrically 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 2 curves, and hence is principally polarizable:

• $y^2+(x^3+a+1)y=ax^6+(a+1)x^4+(a+1)x^3+ax+1$
• $y^2+(x^3+a)y=(a+1)x^6+ax^4+ax^3+(a+1)x+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 7 259 3136 58275 1109227 17172736 267001147 4324529475 69244764736 1100771362579

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 18 47 226 1082 4191 16298 65986 264143 1049778

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{-7})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$
All geometric endomorphisms are defined over $\F_{2^{6}}$.

## 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.ab_ab $\F_{2}$ 2.2.b_ab

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.d_f $2$ 2.16.b_ap 2.4.g_r $3$ 2.64.as_ib 2.4.ag_r $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.d_f $2$ 2.16.b_ap 2.4.g_r $3$ 2.64.as_ib 2.4.ag_r $6$ (not in LMFDB) 2.4.a_ab $6$ (not in LMFDB) 2.4.a_b $12$ (not in LMFDB)