Properties

Label 3.11.ap_ed_ari
Base Field $\F_{11}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{11}$
Dimension:  $3$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 - 5 x + 11 x^{2} )( 1 - 4 x + 11 x^{2} )$
Frobenius angles:  $\pm0.140218899004$, $\pm0.228229222880$, $\pm0.293962833700$
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
$A(\F_{q^r})$ 336 1645056 2523931200 3244050432000 4210466656559856 5564432369986560000 7399286919470169340176 9849227444670893948928000 13110067747977003449539588800 17449543701078556036959822260736

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 111 1422 15127 162327 1773000 19484637 214347887 2357960922 25937634831

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 1.11.af $\times$ 1.11.ae 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 $\F_{11}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ah_t_abi$2$(not in LMFDB)
3.11.af_h_k$2$(not in LMFDB)
3.11.ad_ab_cc$2$(not in LMFDB)
3.11.d_ab_acc$2$(not in LMFDB)
3.11.f_h_ak$2$(not in LMFDB)
3.11.h_t_bi$2$(not in LMFDB)
3.11.p_ed_ri$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ah_t_abi$2$(not in LMFDB)
3.11.af_h_k$2$(not in LMFDB)
3.11.ad_ab_cc$2$(not in LMFDB)
3.11.d_ab_acc$2$(not in LMFDB)
3.11.f_h_ak$2$(not in LMFDB)
3.11.h_t_bi$2$(not in LMFDB)
3.11.p_ed_ri$2$(not in LMFDB)