Properties

Label 3.13.ar_fe_axs
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:  $( 1 - 6 x + 13 x^{2} )( 1 - 11 x + 55 x^{2} - 143 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.129998747777$, $\pm0.187167041811$, $\pm0.292104599859$
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
$A(\F_{q^r})$ 568 4305440 10967735224 23752337500800 51448306585454208 112556148246120910880 247097698041534970160056 542820157999655317229836800 1192549670586296126218272874168 2620004638956813938851726677811200

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 149 2271 29113 373192 4831133 62756943 815759857 10604645013 137858965164

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ag $\times$ 2.13.al_cd 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.af_c_bs$2$(not in LMFDB)
3.13.f_c_abs$2$(not in LMFDB)
3.13.r_fe_xs$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.af_c_bs$2$(not in LMFDB)
3.13.f_c_abs$2$(not in LMFDB)
3.13.r_fe_xs$2$(not in LMFDB)
3.13.ap_ei_atm$4$(not in LMFDB)
3.13.ah_y_aco$4$(not in LMFDB)
3.13.h_y_co$4$(not in LMFDB)
3.13.p_ei_tm$4$(not in LMFDB)