# Properties

 Label 2.4.ad_h 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: $1 - 3 x + 7 x^{2} - 12 x^{3} + 16 x^{4}$ Frobenius angles: $\pm0.190783854037$, $\pm0.524117187371$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{13})$$ Galois group: $C_2^2$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

• $y^2+(x^3+x+1)y=(a+1)x^6+(a+1)x^5+(a+1)x^4+(a+1)x^3+(a+1)x^2+(a+1)x+a+1$
• $y^2+(x^3+x+1)y=(a+1)x^6+ax^5+ax^4+ax^3+(a+1)x^2+ax+a+1$
• $y^2+(x^3+x+1)y=(a+1)x^6+(a+1)x^5+(a+1)x^4+ax^3+(a+1)x^2+ax+a$
• $y^2+(x^3+x+1)y=(a+1)x^6+ax^5+ax^4+(a+1)x^3+(a+1)x^2+(a+1)x+a$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 9 351 4212 63531 1141299 17740944 268402689 4264772499 68719584492 1099012730751

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 22 65 250 1112 4327 16382 65074 262145 1048102

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{13})$$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.el 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-39})$$$)$
All geometric endomorphisms are defined over $\F_{2^{12}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 2.16.f_j and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{13})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 2.64.a_el and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{13})$$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.d_h $2$ 2.16.f_j 2.4.a_af $3$ 2.64.a_el 2.4.d_h $3$ 2.64.a_el
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.d_h $2$ 2.16.f_j 2.4.a_af $3$ 2.64.a_el 2.4.d_h $3$ 2.64.a_el 2.4.a_f $12$ (not in LMFDB)