Properties

Label 4.2.ad_c_a_b
Base Field $\F_{2}$
Dimension $4$
Ordinary No
$p$-rank $4$
Principally polarizable Yes

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Invariants

Base field:  $\F_{2}$
Dimension:  $4$
Weil polynomial:  $1 - 3 x + 2 x^{2} + x^{4} + 8 x^{6} - 24 x^{7} + 16 x^{8}$
Frobenius angles:  $\pm0.0298810195513$, $\pm0.106143893905$, $\pm0.506143893905$, $\pm0.829881019551$
Angle rank:  $2$ (numerical)
Number field:  8.0.26265625.1
Galois group:  $C_2^2:C_4$

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1 55 1621 27775 609961 24161005 223259821 4066287775 72597199861 1153207515625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 0 0 4 15 90 105 244 540 1075

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is 8.0.26265625.1.
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{5}}$ is 2.32.aj_cb 2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.1025.1$)$
All geometric endomorphisms are defined over $\F_{2^{5}}$.

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.d_c_a_b$2$4.4.af_g_u_adb
4.2.c_c_f_l$5$(not in LMFDB)
4.2.c_h_k_v$5$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.2.d_c_a_b$2$4.4.af_g_u_adb
4.2.c_c_f_l$5$(not in LMFDB)
4.2.c_h_k_v$5$(not in LMFDB)
4.2.ac_c_af_l$10$(not in LMFDB)
4.2.ac_h_ak_v$10$(not in LMFDB)
4.2.a_f_a_n$10$(not in LMFDB)
4.2.ab_ac_b_d$15$(not in LMFDB)
4.2.a_af_a_n$20$(not in LMFDB)
4.2.b_ac_ab_d$30$(not in LMFDB)

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. This example was found by Stirpe in 2014 [10.1016/j.jnt.2014.02.016, MR:3227356], refuting a claim made in 1975 by Leitzel-Madan-Queen [0.1016/0022-314X(75)90004-9, MR:0369326].