Properties

Label 3.13.ap_ef_ass
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:  $( 1 - 6 x + 13 x^{2} )( 1 - 9 x + 42 x^{2} - 117 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.136139978944$, $\pm0.187167041811$, $\pm0.390198274089$
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
$A(\F_{q^r})$ 688 4650880 10964218432 23480618803200 51274688479675888 112587197950684487680 247202755971073843853488 542872893304061401979289600 1192537349150551687055542497088 2619975166112142019158876065814400

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 163 2270 28783 371939 4832464 62783615 815839103 10604535446 137857414363

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ag $\times$ 2.13.aj_bq 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_b_s$2$(not in LMFDB)
3.13.d_b_as$2$(not in LMFDB)
3.13.p_ef_ss$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ad_b_s$2$(not in LMFDB)
3.13.d_b_as$2$(not in LMFDB)
3.13.p_ef_ss$2$(not in LMFDB)
3.13.an_dn_apm$4$(not in LMFDB)
3.13.af_t_aco$4$(not in LMFDB)
3.13.f_t_co$4$(not in LMFDB)
3.13.n_dn_pm$4$(not in LMFDB)