Properties

Label 3.7.am_cp_aip
Base field $\F_{7}$
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:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $( 1 - 5 x + 7 x^{2} )( 1 - 7 x + 25 x^{2} - 49 x^{3} + 49 x^{4} )$
  $1 - 12 x + 67 x^{2} - 223 x^{3} + 469 x^{4} - 588 x^{5} + 343 x^{6}$
Frobenius angles:  $\pm0.106147807505$, $\pm0.162349854003$, $\pm0.351370772325$
Angle rank:  $3$ (numerical)
Isomorphism classes:  1

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $57$ $97071$ $42224004$ $14146059759$ $4762715284272$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-4$ $40$ $359$ $2452$ $16861$ $117973$ $826220$ $5775892$ $40378823$ $282494075$

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 $\F_{7}$.

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 2.7.ah_z and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

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.ac_ad_bb$2$(not in LMFDB)
3.7.c_ad_abb$2$(not in LMFDB)
3.7.m_cp_ip$2$(not in LMFDB)
3.7.ag_z_acv$3$(not in LMFDB)
3.7.ad_e_c$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.7.ac_ad_bb$2$(not in LMFDB)
3.7.c_ad_abb$2$(not in LMFDB)
3.7.m_cp_ip$2$(not in LMFDB)
3.7.ag_z_acv$3$(not in LMFDB)
3.7.ad_e_c$3$(not in LMFDB)
3.7.al_ci_ahq$6$(not in LMFDB)
3.7.ai_bn_aet$6$(not in LMFDB)
3.7.d_e_ac$6$(not in LMFDB)
3.7.g_z_cv$6$(not in LMFDB)
3.7.i_bn_et$6$(not in LMFDB)
3.7.l_ci_hq$6$(not in LMFDB)