# Properties

 Label 3.16.au_gv_abiu Base Field $\F_{2^{4}}$ Dimension $3$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $3$ L-polynomial: $( 1 - 4 x )^{2}( 1 - 12 x + 65 x^{2} - 192 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.0826163580681$, $\pm0.320878822416$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

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

## 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 $A(\F_{q^r})$ 1062 13965300 67160711142 279547543828800 1149343372460251782 4717788853900646733300 19339337645703160662929238 79227421077482840321872876800 324520116587361235231008975262998 1329229636192222503973373548304032500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 211 4005 65087 1045317 16760947 268387221 4294927103 68719807701 1099512984691

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.am_cn and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.16.ai : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.am_cn : 4.0.27792.2.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

## 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 3.16.ae_ap_fg $2$ (not in LMFDB) 3.16.e_ap_afg $2$ (not in LMFDB) 3.16.u_gv_biu $2$ (not in LMFDB) 3.16.ai_bh_aeu $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ae_ap_fg $2$ (not in LMFDB) 3.16.e_ap_afg $2$ (not in LMFDB) 3.16.u_gv_biu $2$ (not in LMFDB) 3.16.ai_bh_aeu $3$ (not in LMFDB) 3.16.am_dd_aou $4$ (not in LMFDB) 3.16.m_dd_ou $4$ (not in LMFDB) 3.16.aq_ez_ayu $6$ (not in LMFDB) 3.16.i_bh_eu $6$ (not in LMFDB) 3.16.q_ez_yu $6$ (not in LMFDB)