# Properties

 Label 3.11.ap_dz_aqk 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: $( 1 - 6 x + 11 x^{2} )( 1 - 9 x + 38 x^{2} - 99 x^{3} + 121 x^{4} )$ Frobenius angles: $\pm0.0468428922585$, $\pm0.140218899004$, $\pm0.380176225592$ Angle rank: $2$ (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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 312 1505088 2325611808 3102311467008 4154340757959912 5556647549495577600 7403498659859443910808 9851799344685116613230592 13110361394916624828365551968 17449270344778098774861829700928

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 103 1314 14471 160167 1770520 19495725 214403855 2358013734 25937228503

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.ag $\times$ 2.11.aj_bm and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{11}$
 The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.acna 2 $\times$ 1.1771561.dly. The endomorphism algebra for each factor is: 1.1771561.acna 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-51})$$$)$ 1.1771561.dly : $$\Q(\sqrt{-2})$$.
All geometric endomorphisms are defined over $\F_{11^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{11^{2}}$  The base change of $A$ to $\F_{11^{2}}$ is 1.121.ao $\times$ 2.121.af_ads. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{11^{3}}$  The base change of $A$ to $\F_{11^{3}}$ is 1.1331.as $\times$ 2.1331.a_acna. The endomorphism algebra for each factor is:

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 3.11.ad_af_be $2$ (not in LMFDB) 3.11.d_af_abe $2$ (not in LMFDB) 3.11.p_dz_qk $2$ (not in LMFDB) 3.11.ag_q_abe $3$ (not in LMFDB) 3.11.d_af_abe $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.11.ad_af_be $2$ (not in LMFDB) 3.11.d_af_abe $2$ (not in LMFDB) 3.11.p_dz_qk $2$ (not in LMFDB) 3.11.ag_q_abe $3$ (not in LMFDB) 3.11.d_af_abe $3$ (not in LMFDB) 3.11.ag_q_abe $6$ (not in LMFDB) 3.11.ad_af_be $6$ (not in LMFDB) 3.11.g_q_be $6$ (not in LMFDB) 3.11.ag_g_be $12$ (not in LMFDB) 3.11.g_g_abe $12$ (not in LMFDB)