Properties

Label 3.13.ap_eb_ars
Base field $\F_{13}$
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:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 36 x^{2} - 104 x^{3} + 169 x^{4} )$
  $1 - 15 x + 105 x^{2} - 460 x^{3} + 1365 x^{4} - 2535 x^{5} + 2197 x^{6}$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.147614849952$, $\pm0.431019279425$
Angle rank:  $3$ (numerical)

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $658$ $4394124$ $10465874272$ $23043647504304$ $51092959908644458$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $155$ $2168$ $28247$ $370619$ $4831592$ $62782103$ $815808767$ $10604532104$ $137858690675$

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 $\F_{13}$.

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bk 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
3.13.ab_ah_bs$2$(not in LMFDB)
3.13.b_ah_abs$2$(not in LMFDB)
3.13.p_eb_rs$2$(not in LMFDB)
3.13.ag_bh_afg$3$(not in LMFDB)
3.13.ad_j_abc$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.13.ab_ah_bs$2$(not in LMFDB)
3.13.b_ah_abs$2$(not in LMFDB)
3.13.p_eb_rs$2$(not in LMFDB)
3.13.ag_bh_afg$3$(not in LMFDB)
3.13.ad_j_abc$3$(not in LMFDB)
3.13.an_dl_aoy$6$(not in LMFDB)
3.13.ak_cn_aku$6$(not in LMFDB)
3.13.d_j_bc$6$(not in LMFDB)
3.13.g_bh_fg$6$(not in LMFDB)
3.13.k_cn_ku$6$(not in LMFDB)
3.13.n_dl_oy$6$(not in LMFDB)