# Properties

 Label 2.169.abv_bic 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: $1 - 47 x + 886 x^{2} - 7943 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0585395387306$, $\pm0.191425177757$ Angle rank: $2$ (numerical) Number field: 4.0.1413788.1 Galois group: $D_{4}$ Jacobians: 8

This isogeny class is simple and geometrically 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 8 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21458 803344604 23284933090016 665416595154354368 19005003162788553065538 542800869906408642101970176 15502932902552857335103158299282 442779263562039767245167652705465088 12646218551394779377627094698080897673952 361188648080869407150446917304838844580311644

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 123 28125 4824084 815730705 137858777563 23298089394594 3937376411069091 665416608860373153 112455406940080971684 19004963774688110895805

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.1413788.1.
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.bv_bic $2$ (not in LMFDB)