Properties

 Label 2.169.abu_bhf 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: $( 1 - 25 x + 169 x^{2} )( 1 - 21 x + 169 x^{2} )$ Frobenius angles: $\pm0.0885687144757$, $\pm0.200716263733$ Angle rank: $2$ (numerical) Jacobians: 24

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 24 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21605 804678225 23290541556560 665433956969861625 19005047856882235388525 542800972688938165079654400 15502933129809347360408818190765 442779264084298243996746535316531625 12646218552689583593035234652664573603920 361188648084198486761338212463730622692330625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 124 28172 4825246 815751988 137859101764 23298093806222 3937376468786836 665416609645232548 112455406951594906414 19004963774863279849052

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.az $\times$ 1.169.av 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 is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.ae_ahf $2$ (not in LMFDB) 2.169.e_ahf $2$ (not in LMFDB) 2.169.bu_bhf $2$ (not in LMFDB)