Properties

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

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Invariants

Base field:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 40 x^{2} - 104 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.229660330050$, $\pm0.383259645523$
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
$A(\F_{q^r})$ 686 4638732 10926245792 23381009108016 51114847857411366 112421676386927432448 247085646745369705463918 542818548136892226822733824 1192527624141034433696590886432 2619985668964138696373481391519212

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 163 2264 28663 370779 4825360 62753879 815757439 10604448968 137857967003

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bo 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.ab_ad_cu$2$(not in LMFDB)
3.13.b_ad_acu$2$(not in LMFDB)
3.13.p_ef_su$2$(not in LMFDB)
3.13.ag_bl_aey$3$(not in LMFDB)
3.13.ad_n_ai$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ab_ad_cu$2$(not in LMFDB)
3.13.b_ad_acu$2$(not in LMFDB)
3.13.p_ef_su$2$(not in LMFDB)
3.13.ag_bl_aey$3$(not in LMFDB)
3.13.ad_n_ai$3$(not in LMFDB)
3.13.an_dp_aps$6$(not in LMFDB)
3.13.ak_cr_alc$6$(not in LMFDB)
3.13.d_n_i$6$(not in LMFDB)
3.13.g_bl_ey$6$(not in LMFDB)
3.13.k_cr_lc$6$(not in LMFDB)
3.13.n_dp_ps$6$(not in LMFDB)