# Properties

 Label 2.13.ag_s Base Field $\F_{13}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{13}$ Dimension: $2$ L-polynomial: $1 - 6 x + 18 x^{2} - 78 x^{3} + 169 x^{4}$ Frobenius angles: $\pm0.0497783698316$, $\pm0.549778369832$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{17})$$ Galois group: $C_2^2$ Jacobians: 8

This isogeny class is simple but not 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:

• $y^2=8x^6+10x^5+4x^4+2x^3+6x^2+2x+7$
• $y^2=9x^6+10x^5+7x^4+3x^3+5x$
• $y^2=6x^6+6x^5+4x^4+4x^2+7x+6$
• $y^2=8x^6+11x^5+2x^4+2x^2+2x+8$
• $y^2=3x^6+4x^5+8x^4+11x^3+6x^2+3x+6$
• $y^2=7x^6+x^5+7x^4+6x^3+11x^2+6x+8$
• $y^2=12x^6+4x^5+3x^4+5x^3+9x^2+x+7$
• $y^2=11x^6+11x^4+x^3+6x^2+11x+4$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 104 28288 4557800 800210944 137856264104 23298088028800 3936037434568616 665387319878221824 112457206783202925800 19004963775156489689728

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 170 2072 28014 371288 4826810 62727176 815694814 10604669096 137858491850

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{17})$$.
Endomorphism algebra over $\overline{\F}_{13}$
 The base change of $A$ to $\F_{13^{4}}$ is 1.28561.ako 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-17})$$$)$
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{13^{2}}$  The base change of $A$ to $\F_{13^{2}}$ is the simple isogeny class 2.169.a_ako and its endomorphism algebra is $$\Q(i, \sqrt{17})$$.

## 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 2.13.g_s $2$ 2.169.a_ako 2.13.a_ai $8$ (not in LMFDB) 2.13.a_i $8$ (not in LMFDB)