# Properties

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

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## Invariants

 Base field: $\F_{13^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 24 x + 169 x^{2} )^{2}$ Frobenius angles: $\pm0.125665916378$, $\pm0.125665916378$ Angle rank: $1$ (numerical) Jacobians: 9

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 9 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 21316 802248976 23282111123716 665417390653440000 19005044101524556310596 542801092413737557716007696 15502933721134081189770835107076 442779265947739709696484208803840000 12646218557102226267280733954285890174276 361188648091785272621232733845069559875820816

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 122 28086 4823498 815731678 137859074522 23298098945046 3937376618969258 665416612445645758 112455406990833950522 19004963775262480076406

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The isogeny class factors as 1.169.ay 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
All geometric endomorphisms are defined over $\F_{13^{2}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.

 Subfield Primitive Model $\F_{13}$ 2.13.a_ay

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.a_aje $2$ (not in LMFDB) 2.169.bw_bje $2$ (not in LMFDB) 2.169.y_pr $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.169.a_aje $2$ (not in LMFDB) 2.169.bw_bje $2$ (not in LMFDB) 2.169.y_pr $3$ (not in LMFDB) 2.169.abi_wg $4$ (not in LMFDB) 2.169.au_qw $4$ (not in LMFDB) 2.169.ao_du $4$ (not in LMFDB) 2.169.a_je $4$ (not in LMFDB) 2.169.o_du $4$ (not in LMFDB) 2.169.u_qw $4$ (not in LMFDB) 2.169.bi_wg $4$ (not in LMFDB) 2.169.ay_pr $6$ (not in LMFDB) 2.169.a_ajg $8$ (not in LMFDB) 2.169.a_jg $8$ (not in LMFDB) 2.169.ak_acr $12$ (not in LMFDB) 2.169.k_acr $12$ (not in LMFDB)