# Properties

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

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $( 1 - 4 x )^{2}( 1 + 16 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.5$ Angle rank: $0$ (numerical) Jacobians: 4

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 contains the Jacobians of 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 153 65025 16260993 4228250625 1097366239233 281474943156225 72048798481743873 18445618199572250625 4722330454210064678913 1208925819612430151450625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 257 3969 64513 1046529 16777217 268402689 4294705153 68718952449 1099511627777

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 1.16.a 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.a : $$\Q(\sqrt{-1})$$.
Endomorphism algebra over $\overline{\F}_{2^{4}}$
 The base change of $A$ to $\F_{2^{16}}$ is 1.65536.ats 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 $\F_{2^{16}}$.
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.bg. The endomorphism algebra for each factor is: 1.256.abg : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.256.bg : 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.i_bg $2$ 2.256.a_ats 2.16.e_bg $3$ (not in LMFDB) 2.16.aq_ds $4$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.16.i_bg $2$ 2.256.a_ats 2.16.e_bg $3$ (not in LMFDB) 2.16.aq_ds $4$ (not in LMFDB) 2.16.a_abg $4$ (not in LMFDB) 2.16.a_bg $4$ (not in LMFDB) 2.16.q_ds $4$ (not in LMFDB) 2.16.ae_bg $6$ (not in LMFDB) 2.16.a_a $8$ (not in LMFDB) 2.16.am_cm $12$ (not in LMFDB) 2.16.ai_bw $12$ (not in LMFDB) 2.16.ae_a $12$ (not in LMFDB) 2.16.a_aq $12$ (not in LMFDB) 2.16.a_q $12$ (not in LMFDB) 2.16.e_a $12$ (not in LMFDB) 2.16.i_bw $12$ (not in LMFDB) 2.16.m_cm $12$ (not in LMFDB) 2.16.ae_q $20$ (not in LMFDB) 2.16.e_q $20$ (not in LMFDB)