Properties

 Label 2.169.abv_bie Base Field $\F_{13^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{13^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 25 x + 169 x^{2} )( 1 - 22 x + 169 x^{2} )$ Frobenius angles: $\pm0.0885687144757$, $\pm0.178912375022$ Angle rank: $2$ (numerical) Jacobians: 22

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 contains the Jacobians of 22 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21460 803462400 23286296559760 665425223211340800 19005042298295333077300 542801011479319554180710400 15502933331308488202467838412980 442779264670789479291298574632243200 12646218553837734126790967833290057669520 361188648085305602814271438059249622659360000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 123 28129 4824366 815741281 137859061443 23298095471182 3937376519962827 665416610526622081 112455406961804735934 19004963774921533896049

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.az $\times$ 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:
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.ad_aie $2$ (not in LMFDB) 2.169.d_aie $2$ (not in LMFDB) 2.169.bv_bie $2$ (not in LMFDB) 2.169.aba_nz $3$ (not in LMFDB) 2.169.ac_ajd $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.ad_aie $2$ (not in LMFDB) 2.169.d_aie $2$ (not in LMFDB) 2.169.bv_bie $2$ (not in LMFDB) 2.169.aba_nz $3$ (not in LMFDB) 2.169.ac_ajd $3$ (not in LMFDB) 2.169.abw_bjd $6$ (not in LMFDB) 2.169.ay_mb $6$ (not in LMFDB) 2.169.c_ajd $6$ (not in LMFDB) 2.169.y_mb $6$ (not in LMFDB) 2.169.ba_nz $6$ (not in LMFDB) 2.169.bw_bjd $6$ (not in LMFDB)