# Properties

 Label 2.16.am_cm Base Field $\F_{2^{4}}$ Dimension $2$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $( 1 - 4 x )^{2}( 1 - 4 x + 16 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.333333333333$ Angle rank: $0$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2, 1/2, 1/2]$

## 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 $A(\F_{q^r})$ 117 61425 16769025 4278189825 1096294593537 281200199450625 72044400972087297 18446462598732775425 4722366482732206260225 1208924666693124566810625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 241 4097 65281 1045505 16760833 268386305 4294901761 68719476737 1099510579201

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 1.16.ae 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 $\infty$. 1.16.ae : $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{2^{4}}$
 The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{24}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{8}}$  The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg $\times$ 1.256.q. The endomorphism algebra for each factor is: 1.256.abg : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.256.q : $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{2^{12}}$  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.ey. The endomorphism algebra for each factor is: 1.4096.aey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.4096.ey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.

## 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 2.16.ae_a $2$ 2.256.aq_a 2.16.e_a $2$ 2.256.aq_a 2.16.m_cm $2$ 2.256.aq_a 2.16.a_abg $3$ (not in LMFDB) 2.16.a_q $3$ (not in LMFDB) 2.16.m_cm $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.ae_a $2$ 2.256.aq_a 2.16.e_a $2$ 2.256.aq_a 2.16.m_cm $2$ 2.256.aq_a 2.16.a_abg $3$ (not in LMFDB) 2.16.a_q $3$ (not in LMFDB) 2.16.m_cm $3$ (not in LMFDB) 2.16.ae_bg $4$ (not in LMFDB) 2.16.e_bg $4$ (not in LMFDB) 2.16.aq_ds $6$ (not in LMFDB) 2.16.ai_bw $6$ (not in LMFDB) 2.16.i_bw $6$ (not in LMFDB) 2.16.q_ds $6$ (not in LMFDB) 2.16.ai_bg $12$ (not in LMFDB) 2.16.a_aq $12$ (not in LMFDB) 2.16.a_bg $12$ (not in LMFDB) 2.16.i_bg $12$ (not in LMFDB) 2.16.a_a $24$ (not in LMFDB) 2.16.ae_q $30$ (not in LMFDB) 2.16.e_q $30$ (not in LMFDB)