Properties

Label 3.7.ak_cc_agu
Base Field $\F_{7}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x + 7 x^{2} )( 1 - 3 x + 7 x^{2} )^{2}$
Frobenius angles:  $\pm0.227185525829$, $\pm0.308124534521$, $\pm0.308124534521$
Angle rank:  $2$ (numerical)

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
$A(\F_{q^r})$ 100 145200 52561600 15289560000 4796192252500 1615529973657600 556037840198563900 191412267980899680000 65723180397352829915200 22542768937234848538830000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 58 436 2642 16978 116716 819838 5759714 40360252 282518218

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.ae $\times$ 1.7.ad 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{7}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.ae_m_au$2$(not in LMFDB)
3.7.ac_g_i$2$(not in LMFDB)
3.7.c_g_ai$2$(not in LMFDB)
3.7.e_m_u$2$(not in LMFDB)
3.7.k_cc_gu$2$(not in LMFDB)
3.7.ah_bk_aed$3$(not in LMFDB)
3.7.ab_ad_bi$3$(not in LMFDB)
3.7.ab_a_bf$3$(not in LMFDB)
3.7.c_g_bo$3$(not in LMFDB)
3.7.i_y_ca$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.ae_m_au$2$(not in LMFDB)
3.7.ac_g_i$2$(not in LMFDB)
3.7.c_g_ai$2$(not in LMFDB)
3.7.e_m_u$2$(not in LMFDB)
3.7.k_cc_gu$2$(not in LMFDB)
3.7.ah_bk_aed$3$(not in LMFDB)
3.7.ab_ad_bi$3$(not in LMFDB)
3.7.ab_a_bf$3$(not in LMFDB)
3.7.c_g_bo$3$(not in LMFDB)
3.7.i_y_ca$3$(not in LMFDB)
3.7.ae_c_u$4$(not in LMFDB)
3.7.e_c_au$4$(not in LMFDB)
3.7.al_ci_ahr$6$(not in LMFDB)
3.7.ai_y_aca$6$(not in LMFDB)
3.7.ah_v_aby$6$(not in LMFDB)
3.7.af_m_az$6$(not in LMFDB)
3.7.af_y_acj$6$(not in LMFDB)
3.7.ae_m_abs$6$(not in LMFDB)
3.7.ac_ag_bg$6$(not in LMFDB)
3.7.ac_g_abo$6$(not in LMFDB)
3.7.ab_m_af$6$(not in LMFDB)
3.7.b_ad_abi$6$(not in LMFDB)
3.7.b_a_abf$6$(not in LMFDB)
3.7.b_m_f$6$(not in LMFDB)
3.7.c_ag_abg$6$(not in LMFDB)
3.7.e_m_bs$6$(not in LMFDB)
3.7.f_m_z$6$(not in LMFDB)
3.7.f_y_cj$6$(not in LMFDB)
3.7.h_v_by$6$(not in LMFDB)
3.7.h_bk_ed$6$(not in LMFDB)
3.7.l_ci_hr$6$(not in LMFDB)
3.7.af_c_z$12$(not in LMFDB)
3.7.ab_c_f$12$(not in LMFDB)
3.7.b_c_af$12$(not in LMFDB)
3.7.f_c_az$12$(not in LMFDB)