# Properties

 Label 2.4.ab_ae Base Field $\F_{2^{2}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 2 x )^{2}( 1 + 3 x + 4 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.769946543837$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 8 144 2744 64800 930248 16447536 265676888 4232347200 68712424424 1096109357904

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 8 40 256 904 4016 16216 64576 262120 1045328

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ae $\times$ 1.4.d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.4.ae : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.4.d : $$\Q(\sqrt{-7})$$.
All geometric endomorphisms are defined over $\F_{2^{2}}$.

## 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.ah_u $2$ 2.16.aj_bo 2.4.b_ae $2$ 2.16.aj_bo 2.4.h_u $2$ 2.16.aj_bo 2.4.f_o $3$ 2.64.az_km
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.ah_u $2$ 2.16.aj_bo 2.4.b_ae $2$ 2.16.aj_bo 2.4.h_u $2$ 2.16.aj_bo 2.4.f_o $3$ 2.64.az_km 2.4.ad_i $4$ 2.256.ab_asm 2.4.d_i $4$ 2.256.ab_asm 2.4.af_o $6$ (not in LMFDB) 2.4.ab_c $6$ (not in LMFDB) 2.4.b_c $6$ (not in LMFDB)