# Properties

 Label 3.13.aq_es_avi Base Field $\F_{13}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{13}$ Dimension: $3$ L-polynomial: $( 1 - 6 x + 13 x^{2} )( 1 - 10 x + 49 x^{2} - 130 x^{3} + 169 x^{4} )$ Frobenius angles: $\pm0.151058869957$, $\pm0.187167041811$, $\pm0.334339837461$ 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 632 4537760 11092755296 23720595868800 51385281922411832 112545827109430561280 247130953553319137008856 542851068350390788895040000 1192553650942543655591798074592 2619988859435131899903417554224800

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 158 2296 29074 372738 4830692 62765386 815806306 10604680408 137858134878

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ag $\times$ 2.13.ak_bx 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.
 Twist Extension Degree Common base change 3.13.ae_c_bi $2$ (not in LMFDB) 3.13.e_c_abi $2$ (not in LMFDB) 3.13.q_es_vi $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ae_c_bi $2$ (not in LMFDB) 3.13.e_c_abi $2$ (not in LMFDB) 3.13.q_es_vi $2$ (not in LMFDB) 3.13.ao_dy_aro $4$ (not in LMFDB) 3.13.ag_w_acm $4$ (not in LMFDB) 3.13.g_w_cm $4$ (not in LMFDB) 3.13.o_dy_ro $4$ (not in LMFDB)