Properties

 Label 5.2.ag_q_ay_w_au Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $0$ Contains a Jacobian No

Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 8 x^{5} + 24 x^{6} - 32 x^{7} + 16 x^{8} )$ Frobenius angles: $\pm0.0377785699724$, $\pm0.148391828106$, $\pm0.250000000000$, $\pm0.398391828106$, $\pm0.787778569972$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

Newton polygon

 $p$-rank: $0$ Slopes: $[1/4, 1/4, 1/4, 1/4, 1/2, 1/2, 3/4, 3/4, 3/4, 3/4]$

Point counts

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1 485 34957 1690225 18761641 1237652585 36577681561 1028469798225 35261765927713 979140494205425

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 1 9 25 17 73 137 241 513 881

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 4.2.ae_g_ae_c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg $\times$ 4.256.q_ds_glk_jitk. The endomorphism algebra for each factor is:
• 1.256.abg : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
• 4.256.q_ds_glk_jitk : the quaternion algebra over 4.0.1088.2 with the following ramification data at primes above $2$, and unramified at all archimedean places:
 $v$ ($2$,$$\pi$$) ($2$,$$\pi + 1$$) $\operatorname{inv}_v$ $1/2$ $1/2$
where $\pi$ is a root of $x^{4} - 2x^{3} + 5x^{2} - 4x + 2$.\n
All geometric endomorphisms are defined over $\F_{2^{8}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 4.4.ae_i_ai_e. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 4.16.a_i_a_q. The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 4.16.a_i_a_q : 8.0.18939904.2.

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 5.2.ac_a_a_g_am $2$ (not in LMFDB) 5.2.c_a_a_g_m $2$ (not in LMFDB) 5.2.g_q_y_w_u $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ac_a_a_g_am $2$ (not in LMFDB) 5.2.c_a_a_g_m $2$ (not in LMFDB) 5.2.g_q_y_w_u $2$ (not in LMFDB) 5.2.ac_e_ai_o_au $4$ (not in LMFDB) 5.2.ac_e_a_ac_m $4$ (not in LMFDB) 5.2.c_e_a_ac_am $4$ (not in LMFDB) 5.2.c_e_i_o_u $4$ (not in LMFDB) 5.2.ae_i_am_o_aq $8$ (not in LMFDB) 5.2.a_e_ae_g_aq $8$ (not in LMFDB) 5.2.a_e_e_g_q $8$ (not in LMFDB) 5.2.e_i_m_o_q $8$ (not in LMFDB) 5.2.ac_ac_i_c_au $16$ (not in LMFDB) 5.2.ac_g_ai_s_au $16$ (not in LMFDB) 5.2.a_ac_a_c_a $16$ (not in LMFDB) 5.2.a_g_a_s_a $16$ (not in LMFDB) 5.2.c_ac_ai_c_u $16$ (not in LMFDB) 5.2.c_g_i_s_u $16$ (not in LMFDB)