# Properties

 Label 2.169.abs_bfe 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: $1 - 44 x + 810 x^{2} - 7436 x^{3} + 28561 x^{4}$ Frobenius angles: $\pm0.0647393693303$, $\pm0.247372633299$ Angle rank: $2$ (numerical) Number field: 4.0.330048.1 Galois group: $D_{4}$ Jacobians: 56

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 56 curves, and hence is principally polarizable:

• $y^2=(8a+3)x^6+(8a+10)x^5+9x^4+3x^3+(5a+12)x^2+(a+11)x+2a+11$
• $y^2=(10a+5)x^6+(5a+7)x^5+(7a+2)x^4+3x^3+(4a+3)x^2+(4a+4)x+9a+9$
• $y^2=(4a+2)x^6+(12a+12)x^5+(6a+6)x^4+(12a+4)x^3+(10a+2)x^2+(9a+12)x+11a+2$
• $y^2=8ax^6+(8a+10)x^5+(a+1)x^4+(7a+8)x^3+8ax^2+(a+11)x+9a$
• $y^2=(9a+6)x^6+(4a+3)x^5+(12a+11)x^4+(4a+7)x^3+(11a+6)x^2+7x+8a+1$
• $y^2=(12a+6)x^6+(8a+9)x^5+(a+9)x^4+(7a+4)x^3+(3a+9)x^2+9ax+5a+4$
• $y^2=(9a+6)x^6+(3a+1)x^5+(11a+12)x^4+(12a+6)x^3+(a+6)x^2+(5a+10)x+9a+2$
• $y^2=(12a+8)x^6+(2a+2)x^5+(7a+6)x^4+(6a+3)x^3+(6a+5)x^2+(11a+10)x+5a+7$
• $y^2=(9a+3)x^6+(5a+9)x^5+(a+5)x^4+(11a+12)x^3+(11a+1)x^2+(7a+3)x+10a+3$
• $y^2=(a+7)x^6+(9a+10)x^5+7x^4+(12a+8)x^3+9ax^2+(2a+7)x+3a+6$
• $y^2=(2a+9)x^6+(11a+4)x^5+8x^4+(7a+9)x^3+(12a+12)x^2+(8a+9)x+7a+3$
• $y^2=(7a+4)x^6+(11a+11)x^5+(7a+5)x^4+(2a+4)x^3+(2a+9)x^2+(7a+3)x+a+9$
• $y^2=(11a+2)x^6+5x^5+(5a+8)x^4+(9a+6)x^3+(11a+1)x^2+(4a+3)x+2a+11$
• $y^2=(3a+5)x^6+(12a+6)x^5+(4a+1)x^4+(7a+1)x^3+(4a+4)x^2+(9a+1)x+6a+6$
• $y^2=(5a+8)x^6+(2a+8)x^5+(8a+3)x^4+(2a+7)x^3+(2a+8)x^2+3x+a+7$
• $y^2=(a+11)x^6+(12a+3)x^5+(7a+1)x^4+12ax^3+(a+7)x^2+(9a+8)x+5a+3$
• $y^2=(11a+7)x^6+(10a+12)x^5+(3a+2)x^4+(11a+7)x^3+(3a+1)x^2+10x+2a+6$
• $y^2=x^6+9x^5+(12a+10)x^4+(8a+2)x^3+(3a+12)x^2+(8a+1)x+3$
• $y^2=(10a+5)x^6+(4a+5)x^5+(10a+10)x^4+(11a+6)x^3+(12a+9)x^2+(5a+7)x+2a+12$
• $y^2=2ax^6+ax^5+2ax^4+12ax^3+6ax^2+4ax+8a$
• and 36 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21892 806763984 23295322932388 665431170999309312 19004985241716052615972 542800704283991919221670864 15502932401594143951032876344836 442779262754672119686378121461055488 12646218551535717014302389853296563681476 361188648086440020414505010982983670669522384

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 126 28246 4826238 815748574 137858647566 23298082285750 3937376283837486 665416607647046590 112455406941334247454 19004963774981224491286

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.330048.1.
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.bs_bfe $2$ (not in LMFDB)