Properties

Label 3.13.ar_fc_axf
Base Field $\F_{13}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 - 10 x + 49 x^{2} - 130 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.151058869957$, $\pm0.334339837461$
Angle rank:  $3$ (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
$A(\F_{q^r})$ 553 4169067 10652248768 23367257826171 51161662525287013 112439219264720538624 247103825522123129583457 542864486380853702327156475 1192579648654060719694371545536 2620012488495518196392536516765347

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 145 2208 28645 371117 4826116 62758497 815826469 10604911584 137859378185

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.ak_bx 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 $\F_{13}$.

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.ad_ai_df$2$(not in LMFDB)
3.13.d_ai_adf$2$(not in LMFDB)
3.13.r_fc_xf$2$(not in LMFDB)
3.13.ai_bq_agg$3$(not in LMFDB)
3.13.af_m_ap$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ad_ai_df$2$(not in LMFDB)
3.13.d_ai_adf$2$(not in LMFDB)
3.13.r_fc_xf$2$(not in LMFDB)
3.13.ai_bq_agg$3$(not in LMFDB)
3.13.af_m_ap$3$(not in LMFDB)
3.13.ap_ei_atl$6$(not in LMFDB)
3.13.am_de_anu$6$(not in LMFDB)
3.13.f_m_p$6$(not in LMFDB)
3.13.i_bq_gg$6$(not in LMFDB)
3.13.m_de_nu$6$(not in LMFDB)
3.13.p_ei_tl$6$(not in LMFDB)