Properties

Label 3.3.ag_v_abs
Base field $\F_{3}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 3 x^{2} )^{3}$
  $1 - 6 x + 21 x^{2} - 44 x^{3} + 63 x^{4} - 54 x^{5} + 27 x^{6}$
Frobenius angles:  $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$
Angle rank:  $1$ (numerical)
Jacobians:  $0$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $8$ $1728$ $54872$ $884736$ $14172488$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $16$ $58$ $124$ $238$ $592$ $1930$ $6460$ $20254$ $60496$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3}$.

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-2}) \)$)$

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_f_ae$2$3.9.g_bn_em
3.3.c_f_e$2$3.9.g_bn_em
3.3.g_v_bs$2$3.9.g_bn_em
3.3.a_a_k$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
3.3.ac_f_ae$2$3.9.g_bn_em
3.3.c_f_e$2$3.9.g_bn_em
3.3.g_v_bs$2$3.9.g_bn_em
3.3.a_a_k$3$(not in LMFDB)
3.3.ac_b_e$4$(not in LMFDB)
3.3.c_b_ae$4$(not in LMFDB)
3.3.ae_i_ao$6$(not in LMFDB)
3.3.a_a_ak$6$(not in LMFDB)
3.3.e_i_o$6$(not in LMFDB)
3.3.ag_t_abo$8$(not in LMFDB)
3.3.ac_d_ai$8$(not in LMFDB)
3.3.c_d_i$8$(not in LMFDB)
3.3.g_t_bo$8$(not in LMFDB)