Properties

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

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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.

Point counts of the abelian variety

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

Point counts of the (virtual) curve

$r$ 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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)