# Properties

 Label 4.3.aj_bn_aee_ii Base Field $\F_{3}$ Dimension $4$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{3}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.5$ Angle rank: $0$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

## 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 $A(\F_{q^r})$ 4 5488 614656 48228544 4856212684 377801998336 25559408866492 1876172350124800 150125140011540736 12107841492722430448

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 7 28 91 325 946 2431 6643 19684 58807

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 1.3.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.3.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.3.a : $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad 3 $\times$ 1.9.g. The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 4 and its endomorphism algebra is $\mathrm{M}_{4}($$$\Q(\sqrt{-3})$$$)$

## 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 4.3.ad_d_a_a $2$ (not in LMFDB) 4.3.d_d_a_a $2$ (not in LMFDB) 4.3.j_bn_ee_ii $2$ (not in LMFDB) 4.3.am_co_aii_rr $3$ (not in LMFDB) 4.3.ag_m_a_abb $3$ (not in LMFDB) 4.3.ag_v_acc_ee $3$ (not in LMFDB) 4.3.ad_d_a_a $3$ (not in LMFDB) 4.3.ad_m_abb_cc $3$ (not in LMFDB) 4.3.a_ag_a_bb $3$ (not in LMFDB) 4.3.a_d_a_a $3$ (not in LMFDB) 4.3.a_m_a_cc $3$ (not in LMFDB) 4.3.d_d_a_a $3$ (not in LMFDB) 4.3.d_m_bb_cc $3$ (not in LMFDB) 4.3.g_m_a_abb $3$ (not in LMFDB) 4.3.g_v_cc_ee $3$ (not in LMFDB) 4.3.j_bn_ee_ii $3$ (not in LMFDB) 4.3.m_co_ii_rr $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ad_d_a_a $2$ (not in LMFDB) 4.3.d_d_a_a $2$ (not in LMFDB) 4.3.j_bn_ee_ii $2$ (not in LMFDB) 4.3.am_co_aii_rr $3$ (not in LMFDB) 4.3.ag_m_a_abb $3$ (not in LMFDB) 4.3.ag_v_acc_ee $3$ (not in LMFDB) 4.3.ad_d_a_a $3$ (not in LMFDB) 4.3.ad_m_abb_cc $3$ (not in LMFDB) 4.3.a_ag_a_bb $3$ (not in LMFDB) 4.3.a_d_a_a $3$ (not in LMFDB) 4.3.a_m_a_cc $3$ (not in LMFDB) 4.3.d_d_a_a $3$ (not in LMFDB) 4.3.d_m_bb_cc $3$ (not in LMFDB) 4.3.g_m_a_abb $3$ (not in LMFDB) 4.3.g_v_cc_ee $3$ (not in LMFDB) 4.3.j_bn_ee_ii $3$ (not in LMFDB) 4.3.m_co_ii_rr $3$ (not in LMFDB) 4.3.ad_j_as_bk $4$ (not in LMFDB) 4.3.d_j_s_bk $4$ (not in LMFDB) 4.3.j_bn_ee_ii $6$ (not in LMFDB) 4.3.ad_d_aj_bb $9$ (not in LMFDB) 4.3.ad_d_j_abb $9$ (not in LMFDB) 4.3.a_d_aj_a $9$ (not in LMFDB) 4.3.a_d_j_a $9$ (not in LMFDB) 4.3.d_d_aj_abb $9$ (not in LMFDB) 4.3.d_d_j_bb $9$ (not in LMFDB) 4.3.ag_j_s_acu $12$ (not in LMFDB) 4.3.ag_s_abk_cl $12$ (not in LMFDB) 4.3.ad_a_j_as $12$ (not in LMFDB) 4.3.a_am_a_cc $12$ (not in LMFDB) 4.3.a_aj_a_bk $12$ (not in LMFDB) 4.3.a_ad_a_a $12$ (not in LMFDB) 4.3.a_a_a_as $12$ (not in LMFDB) 4.3.a_a_a_j $12$ (not in LMFDB) 4.3.a_g_a_bb $12$ (not in LMFDB) 4.3.a_j_a_bk $12$ (not in LMFDB) 4.3.d_a_aj_as $12$ (not in LMFDB) 4.3.g_j_as_acu $12$ (not in LMFDB) 4.3.g_s_bk_cl $12$ (not in LMFDB) 4.3.ad_g_aj_j $15$ (not in LMFDB) 4.3.a_ad_a_j $15$ (not in LMFDB) 4.3.d_g_j_j $15$ (not in LMFDB) 4.3.ag_p_as_s $24$ (not in LMFDB) 4.3.ad_g_aj_s $24$ (not in LMFDB) 4.3.a_ag_a_s $24$ (not in LMFDB) 4.3.a_ad_a_s $24$ (not in LMFDB) 4.3.a_a_a_aj $24$ (not in LMFDB) 4.3.a_a_a_s $24$ (not in LMFDB) 4.3.a_d_a_s $24$ (not in LMFDB) 4.3.a_g_a_s $24$ (not in LMFDB) 4.3.d_g_j_s $24$ (not in LMFDB) 4.3.g_p_s_s $24$ (not in LMFDB) 4.3.a_a_a_a $48$ (not in LMFDB) 4.3.a_d_a_j $60$ (not in LMFDB)