Properties

Label 4.2.ae_g_ae_c
Base Field $\F_{2}$
Dimension $4$
Ordinary No
$p$-rank $0$
Contains a Jacobian No

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Invariants

Base field:  $\F_{2}$
Dimension:  $4$
L-polynomial:  $1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 8 x^{5} + 24 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:  $\pm0.0377785699724$, $\pm0.148391828106$, $\pm0.398391828106$, $\pm0.787778569972$
Angle rank:  $2$ (numerical)
Number field:  8.0.18939904.2
Galois group:  $D_4\times C_2$

This isogeny class is simple and geometrically simple.

Newton polygon

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

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 97 2689 67609 457601 19040809 323696297 4570976881 73309284673 955259018737

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 1 5 17 9 73 153 273 545 881

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is 8.0.18939904.2.
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{8}}$ is the simple isogeny class 4.256.q_ds_glk_jitk and its endomorphism algebra is 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
4.2.e_g_e_c$2$4.4.ae_i_ai_e
4.2.a_c_ae_c$4$(not in LMFDB)
4.2.a_c_e_c$4$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.2.e_g_e_c$2$4.4.ae_i_ai_e
4.2.a_c_ae_c$4$(not in LMFDB)
4.2.a_c_e_c$4$(not in LMFDB)
4.2.a_c_ae_c$8$(not in LMFDB)
4.2.a_c_e_c$8$(not in LMFDB)
4.2.e_g_e_c$8$(not in LMFDB)
4.2.a_ae_a_k$16$(not in LMFDB)
4.2.a_e_a_k$16$(not in LMFDB)