Properties

Label 3.3.ag_t_abo
Base Field $\F_{3}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0540867239847$, $\pm0.304086723985$, $\pm0.445913276015$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4 816 23788 443904 11926244 363956400 10295801996 278189309952 7586211740356 207582049122096

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 12 34 68 198 684 2154 6460 19582 59532

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac $\times$ 2.3.ae_i 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 $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{8}}$ is 1.6561.abi 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-2}) \)$)$
All geometric endomorphisms are defined over $\F_{3^{8}}$.
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
3.3.ac_d_ai$2$3.9.c_af_abc
3.3.c_d_i$2$3.9.c_af_abc
3.3.g_t_bo$2$3.9.c_af_abc
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ac_d_ai$2$3.9.c_af_abc
3.3.c_d_i$2$3.9.c_af_abc
3.3.g_t_bo$2$3.9.c_af_abc
3.3.ag_v_abs$8$(not in LMFDB)
3.3.ac_b_e$8$(not in LMFDB)
3.3.ac_f_ae$8$(not in LMFDB)
3.3.c_b_ae$8$(not in LMFDB)
3.3.c_f_e$8$(not in LMFDB)
3.3.g_v_bs$8$(not in LMFDB)
3.3.ae_i_ao$24$(not in LMFDB)
3.3.a_a_ak$24$(not in LMFDB)
3.3.a_a_k$24$(not in LMFDB)
3.3.e_i_o$24$(not in LMFDB)