# Properties

 Label 2.139.abt_bee Base Field $\F_{139}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{139}$ Dimension: $2$ L-polynomial: $( 1 - 23 x + 139 x^{2} )( 1 - 22 x + 139 x^{2} )$ Frobenius angles: $\pm0.0707251543800$, $\pm0.117174211439$ Angle rank: $2$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## 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 $A(\F_{q^r})$ 13806 364561236 7201681820424 139343158506468384 2692448012954737577826 52020882986543132116867776 1005095263314788523069442152786 19419444688735227776345169435114624 375203088685935935743214232068325511464 7249298872294029907724298389988067273296276

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 95 18865 2681570 373272889 51888763925 7212551347186 1002544420941095 139353668097132529 19370159754758019110 2692452204348871439425

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{139}$
 The isogeny class factors as 1.139.ax $\times$ 1.139.aw 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 $\F_{139}$.

## 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 2.139.ab_aiu $2$ (not in LMFDB) 2.139.b_aiu $2$ (not in LMFDB) 2.139.bt_bee $2$ (not in LMFDB) 2.139.ap_eu $3$ (not in LMFDB) 2.139.ag_acw $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.139.ab_aiu $2$ (not in LMFDB) 2.139.b_aiu $2$ (not in LMFDB) 2.139.bt_bee $2$ (not in LMFDB) 2.139.ap_eu $3$ (not in LMFDB) 2.139.ag_acw $3$ (not in LMFDB) 2.139.abm_yg $6$ (not in LMFDB) 2.139.abd_qq $6$ (not in LMFDB) 2.139.g_acw $6$ (not in LMFDB) 2.139.p_eu $6$ (not in LMFDB) 2.139.bd_qq $6$ (not in LMFDB) 2.139.bm_yg $6$ (not in LMFDB)