# Properties

 Label 2.17.ak_by 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: $( 1 - 8 x + 17 x^{2} )( 1 - 2 x + 17 x^{2} )$ Frobenius angles: $\pm0.0779791303774$, $\pm0.422020869623$ Angle rank: $1$ (numerical) Jacobians: 10

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 10 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 160 83200 24088480 6922240000 2011667984800 582622284524800 168396666261788320 48662075833712640000 14063067032705334535840 4064231406646822197280000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 290 4904 82878 1416808 24137570 410384584 6975884158 118587729608 2015993900450

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{17}$
 The isogeny class factors as 1.17.ai $\times$ 1.17.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{17}$
 The base change of $A$ to $\F_{17^{4}}$ is 1.83521.amk 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
All geometric endomorphisms are defined over $\F_{17^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{17^{2}}$  The base change of $A$ to $\F_{17^{2}}$ is 1.289.abe $\times$ 1.289.be. The endomorphism algebra for each factor is:

## 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.ag_s $2$ (not in LMFDB) 2.17.g_s $2$ (not in LMFDB) 2.17.k_by $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.17.ag_s $2$ (not in LMFDB) 2.17.g_s $2$ (not in LMFDB) 2.17.k_by $2$ (not in LMFDB) 2.17.aq_du $4$ (not in LMFDB) 2.17.ae_bm $4$ (not in LMFDB) 2.17.a_abe $4$ (not in LMFDB) 2.17.a_be $4$ (not in LMFDB) 2.17.e_bm $4$ (not in LMFDB) 2.17.q_du $4$ (not in LMFDB) 2.17.a_aq $8$ (not in LMFDB) 2.17.a_q $8$ (not in LMFDB) 2.17.ai_bv $12$ (not in LMFDB) 2.17.ac_an $12$ (not in LMFDB) 2.17.c_an $12$ (not in LMFDB) 2.17.i_bv $12$ (not in LMFDB)