Properties

Label 3.13.ap_ea_arl
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 35 x^{2} - 104 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.126882739163$, $\pm0.439864156467$
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
$A(\F_{q^r})$ 651 4333707 10351962432 22951801980891 51060059851663191 112561965616195243008 247191503987224050100203 542850218228437080259907259 1192544532792050915396807157312 2620011244943064284872767660218787

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 153 2144 28133 370379 4831380 62780759 815805029 10604599328 137859312753

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bj 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_ai_bl$2$(not in LMFDB)
3.13.b_ai_abl$2$(not in LMFDB)
3.13.p_ea_rl$2$(not in LMFDB)
3.13.ag_bg_afi$3$(not in LMFDB)
3.13.ad_i_abh$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ab_ai_bl$2$(not in LMFDB)
3.13.b_ai_abl$2$(not in LMFDB)
3.13.p_ea_rl$2$(not in LMFDB)
3.13.ag_bg_afi$3$(not in LMFDB)
3.13.ad_i_abh$3$(not in LMFDB)
3.13.an_dk_aot$6$(not in LMFDB)
3.13.ak_cm_aks$6$(not in LMFDB)
3.13.d_i_bh$6$(not in LMFDB)
3.13.g_bg_fi$6$(not in LMFDB)
3.13.k_cm_ks$6$(not in LMFDB)
3.13.n_dk_ot$6$(not in LMFDB)