# Properties

 Label 2.169.abw_bja Base Field $\F_{13^{2}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{13^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 13 x )^{2}( 1 - 22 x + 169 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.178912375022$ Angle rank: $1$ (numerical) Jacobians: 20

This isogeny class is not simple.

## Newton polygon $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 20 curves, and hence is principally polarizable:

• $y^2=(2a+1)x^6+(a+6)x^5+(9a+5)x^4+(9a+12)x^3+4ax^2+(11a+9)x+a+12$
• $y^2=(a+5)x^6+(11a+9)x^5+3x^4+12ax^3+(4a+12)x^2+3ax+4a+9$
• $y^2=(10a+8)x^6+(9a+4)x^5+(8a+10)x^4+12ax^3+(10a+1)x^2+3ax+2a+4$
• $y^2=8ax^6+4ax^5+12ax^4+3ax^3+12ax^2+4ax+8a$
• $y^2=(9a+10)x^6+(5a+10)x^5+(6a+12)x^4+(3a+11)x^3+(6a+12)x^2+(5a+10)x+9a+10$
• $y^2=9x^6+3ax^5+(6a+8)x^4+(11a+2)x^3+(6a+8)x^2+3ax+9$
• $y^2=(6a+1)x^6+(5a+3)x^5+(7a+6)x^4+(4a+11)x^3+(7a+6)x^2+(5a+3)x+6a+1$
• $y^2=5ax^6+2ax^4+11ax^3+11ax^2+8a$
• $y^2=(2a+1)x^6+(4a+3)x^5+(5a+3)x^4+(8a+2)x^3+(2a+5)x^2+(a+9)x+10a+11$
• $y^2=(6a+4)x^6+(9a+2)x^5+(9a+4)x^4+(2a+8)x^3+(9a+4)x^2+(9a+2)x+6a+4$
• $y^2=(6a+8)x^6+(5a+10)x^5+(12a+8)x^4+9x^3+(12a+8)x^2+(5a+10)x+6a+8$
• $y^2=(11a+3)x^6+(12a+7)x^5+(4a+1)x^4+(4a+8)x^3+(9a+4)x^2+(2a+7)x+12a$
• $y^2=(7a+8)x^6+(5a+5)x^5+(6a+6)x^4+(11a+4)x^3+(6a+6)x^2+(5a+5)x+7a+8$
• $y^2=(12a+11)x^6+(10a+1)x^5+(4a+11)x^4+10x^3+(4a+11)x^2+(10a+1)x+12a+11$
• $y^2=9ax^6+5ax^5+9ax^4+8ax^3+10ax^2+2ax+4a$
• $y^2=(2a+2)x^6+(5a+11)x^5+(10a+12)x^4+(11a+9)x^3+(9a+10)x^2+4ax+3a+11$
• $y^2=(11a+3)x^6+(9a+7)x^5+(8a+8)x^4+(5a+10)x^3+(11a+7)x^2+(12a+10)x+3a+8$
• $y^2=9ax^6+7ax^5+8ax^4+11ax^2+11ax+9a$
• $y^2=12x^6+(11a+4)x^5+(6a+2)x^4+(2a+11)x^3+(6a+2)x^2+(11a+4)x+12$
• $y^2=(3a+11)x^6+(4a+2)x^5+(3a+1)x^4+(9a+4)x^3+(a+3)x^2+4ax+2a+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21312 802013184 23279325915456 665399220653260800 19004958209563460165952 542800764409177865855287296 15502932655305798311356598344512 442779262923730269141293513519923200 12646218549535878102276395869049620789056 361188648075159905971438065308456186141589504

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 122 28078 4822922 815709406 137858451482 23298084866446 3937376348274218 665416607901110206 112455406923550852538 19004963774387689319278

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.aba $\times$ 1.169.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.169.aba : the quaternion algebra over $$\Q$$ ramified at $13$ and $\infty$. 1.169.aw : $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{13^{2}}$.

## 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 2.169.ae_aja $2$ (not in LMFDB) 2.169.e_aja $2$ (not in LMFDB) 2.169.bw_bja $2$ (not in LMFDB) 2.169.abb_oa $3$ (not in LMFDB) 2.169.ad_aka $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.ae_aja $2$ (not in LMFDB) 2.169.e_aja $2$ (not in LMFDB) 2.169.bw_bja $2$ (not in LMFDB) 2.169.abb_oa $3$ (not in LMFDB) 2.169.ad_aka $3$ (not in LMFDB) 2.169.abx_bka $6$ (not in LMFDB) 2.169.az_ma $6$ (not in LMFDB) 2.169.d_aka $6$ (not in LMFDB) 2.169.z_ma $6$ (not in LMFDB) 2.169.bb_oa $6$ (not in LMFDB) 2.169.bx_bka $6$ (not in LMFDB)