# Properties

 Label 3.3.ae_j_aq Base Field $\F_{3}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - 2 x + 3 x^{2} )( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.116139763599$, $\pm0.304086723985$, $\pm0.616139763599$ Angle rank: $2$ (numerical) Jacobians: 4

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 contains the Jacobians of 4 curves, and hence is principally polarizable:

• $y^2=2x^8+x^7+2x^6+2x^5+2x^2+2$
• $y^2=2x^8+2x^6+2x^4+2$
• $y^2=2x^8+x^7+2x^3+x^2+2$
• $x^4+x^2y^2+2x^2yz+2x^2z^2+2y^4+y^3z+z^4=0$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 8 960 16568 614400 16708648 363833280 10174091992 294794035200 7793263245896 207571534104000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 12 24 92 280 684 2128 6844 20112 59532

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac $\times$ 2.3.ac_c 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 $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac 2 $\times$ 1.81.o. The endomorphism algebra for each factor is: 1.81.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$ 1.81.o : $$\Q(\sqrt{-2})$$.
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.c $\times$ 2.9.a_ac. 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 3.3.a_b_ai $2$ 3.9.c_h_ae 3.3.a_b_i $2$ 3.9.c_h_ae 3.3.e_j_q $2$ 3.9.c_h_ae
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.a_b_ai $2$ 3.9.c_h_ae 3.3.a_b_i $2$ 3.9.c_h_ae 3.3.e_j_q $2$ 3.9.c_h_ae 3.3.ac_ab_i $8$ (not in LMFDB) 3.3.ac_h_ai $8$ (not in LMFDB) 3.3.c_ab_ai $8$ (not in LMFDB) 3.3.c_h_i $8$ (not in LMFDB)