Properties

Label 4.2.ae_e_h_av
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:  $1 - 4 x + 4 x^{2} + 7 x^{3} - 21 x^{4} + 14 x^{5} + 16 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:  $\pm0.0764513550391$, $\pm0.143118021706$, $\pm0.323548644961$, $\pm0.943118021706$
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 31 7471 55831 1343281 14127661 308739901 5138741071 82847868931 1240548212401

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 -3 14 13 39 54 146 301 608 1147

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 $\F_{2^{30}}$ is 1.1073741824.acgor 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{30}}$.
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_e_ah_av$2$4.4.ai_be_acv_ft
4.2.ab_e_ac_j$3$(not in LMFDB)
4.2.f_n_z_bn$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.2.e_e_ah_av$2$4.4.ai_be_acv_ft
4.2.ab_e_ac_j$3$(not in LMFDB)
4.2.f_n_z_bn$3$(not in LMFDB)
4.2.b_e_c_j$5$(not in LMFDB)
4.2.g_t_bq_cr$5$(not in LMFDB)
4.2.af_n_az_bn$6$(not in LMFDB)
4.2.b_e_c_j$6$(not in LMFDB)
4.2.ag_t_abq_cr$10$(not in LMFDB)
4.2.ab_e_ac_j$10$(not in LMFDB)
4.2.a_b_a_ad$10$(not in LMFDB)
4.2.ag_t_abq_cr$15$(not in LMFDB)
4.2.af_n_az_bn$15$(not in LMFDB)
4.2.ad_e_ad_d$15$(not in LMFDB)
4.2.a_ac_a_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.e_e_ah_av$15$(not in LMFDB)
4.2.a_ab_a_ad$20$(not in LMFDB)
4.2.ad_g_aj_n$60$(not in LMFDB)
4.2.a_a_a_h$60$(not in LMFDB)
4.2.a_c_a_j$60$(not in LMFDB)
4.2.d_g_j_n$60$(not in LMFDB)
4.2.a_a_a_ah$120$(not in LMFDB)

Additional information

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