# Properties

 Label 2.17.aj_bs Base Field $\F_{17}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{17}$ Dimension: $2$ L-polynomial: $1 - 9 x + 44 x^{2} - 153 x^{3} + 289 x^{4}$ Frobenius angles: $\pm0.116336544503$, $\pm0.449669877836$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{41})$$ 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:

• $y^2=14x^6+11x^5+2x^4+8x^3+4x^2+14x+13$
• $y^2=5x^6+4x^5+7x^4+3x^3+13x^2+6x$
• $y^2=14x^6+2x^5+15x^4+2x^3+x^2+14x+5$
• $y^2=9x^5+7x^4+10x^3+12x^2+7x+12$
• $y^2=7x^6+6x^5+8x^3+12x^2+10$
• $y^2=11x^6+8x^5+3x^4+8x^3+9x^2+7x+11$
• $y^2=6x^6+15x^5+9x^4+16x^3+6x^2+14x+13$
• $y^2=6x^6+13x^5+16x^4+2x^3+4x^2+16x+12$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 172 85312 24143296 6931770624 2014449490252 582898741743616 168406571598364684 48661978739878425600 14063084451840828627904 4064236334561133871365952

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 297 4914 82993 1418769 24149022 410408721 6975870241 118587876498 2015996344857

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{17}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{41})$$.
Endomorphism algebra over $\overline{\F}_{17}$
 The base change of $A$ to $\F_{17^{6}}$ is 1.24137569.img 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-123})$$$)$
All geometric endomorphisms are defined over $\F_{17^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{17^{2}}$  The base change of $A$ to $\F_{17^{2}}$ is the simple isogeny class 2.289.h_ajg and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{41})$$.
• Endomorphism algebra over $\F_{17^{3}}$  The base change of $A$ to $\F_{17^{3}}$ is the simple isogeny class 2.4913.a_img and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{41})$$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.17.j_bs $2$ (not in LMFDB) 2.17.a_ah $3$ (not in LMFDB) 2.17.j_bs $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.17.j_bs $2$ (not in LMFDB) 2.17.a_ah $3$ (not in LMFDB) 2.17.j_bs $3$ (not in LMFDB) 2.17.a_h $12$ (not in LMFDB)