Properties

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

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Invariants

Base field:  $\F_{2}$
Dimension:  $4$
L-polynomial:  $1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 50 x^{5} + 52 x^{6} - 40 x^{7} + 16 x^{8}$
Frobenius angles:  $\pm0.00978468837242$, $\pm0.190215311628$, $\pm0.409784688372$, $\pm0.609784688372$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{15})\)
Galois group:  $C_4\times C_2$

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 not principally polarizable, and therefore does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1 241 1891 43621 929296 14127661 256112011 4581295525 57170567071 863591055616

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 6 4 10 33 54 124 274 418 781

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{15})\).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{10}}$ is 1.1024.acj 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{10}}$.
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.f_n_z_bn$2$4.4.b_ad_ah_f
4.2.b_e_c_j$3$(not in LMFDB)
4.2.e_e_ah_av$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.2.f_n_z_bn$2$4.4.b_ad_ah_f
4.2.b_e_c_j$3$(not in LMFDB)
4.2.e_e_ah_av$3$(not in LMFDB)
4.2.a_ac_a_j$5$(not in LMFDB)
4.2.f_n_z_bn$5$(not in LMFDB)
4.2.ae_e_h_av$6$(not in LMFDB)
4.2.ab_e_ac_j$6$(not in LMFDB)
4.2.ag_t_abq_cr$15$(not in LMFDB)
4.2.ae_e_h_av$15$(not in LMFDB)
4.2.ad_e_ad_d$15$(not in LMFDB)
4.2.ab_e_ac_j$15$(not in LMFDB)
4.2.a_b_a_ad$15$(not in LMFDB)
4.2.d_e_d_d$15$(not in LMFDB)
4.2.g_t_bq_cr$15$(not in LMFDB)
4.2.a_a_a_h$20$(not in LMFDB)
4.2.a_c_a_j$20$(not in LMFDB)
4.2.a_a_a_ah$40$(not in LMFDB)
4.2.ad_g_aj_n$60$(not in LMFDB)
4.2.a_ab_a_ad$60$(not in LMFDB)
4.2.d_g_j_n$60$(not in LMFDB)

Additional information

This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.