# Properties

 Label 2.4.ad_e 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: $( 1 - 2 x )^{2}( 1 + x + 4 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.580430623255$ 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 $A(\F_{q^r})$ 6 216 2646 54000 1043646 16288776 260225286 4276044000 68655500046 1095615396216

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 16 38 208 1022 3976 15878 65248 261902 1044856

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
 The isogeny class factors as 1.4.ae $\times$ 1.4.b 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.b : $$\Q(\sqrt{-15})$$.
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.af_m $2$ 2.16.ab_ay 2.4.d_e $2$ 2.16.ab_ay 2.4.f_m $2$ 2.16.ab_ay 2.4.d_k $3$ 2.64.abb_ls
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.4.af_m $2$ 2.16.ab_ay 2.4.d_e $2$ 2.16.ab_ay 2.4.f_m $2$ 2.16.ab_ay 2.4.d_k $3$ 2.64.abb_ls 2.4.ab_i $4$ 2.256.abx_boq 2.4.b_i $4$ 2.256.abx_boq 2.4.ad_k $6$ (not in LMFDB) 2.4.ab_g $6$ (not in LMFDB) 2.4.b_g $6$ (not in LMFDB)