# Properties

 Label 2.17.ai_bg 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: $1 - 8 x + 32 x^{2} - 136 x^{3} + 289 x^{4}$ Frobenius angles: $\pm0.00936746300954$, $\pm0.509367463010$ Angle rank: $1$ (numerical) Number field: $$\Q(\zeta_{8})$$ Galois group: $C_2^2$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2=3x^5+8x$
• $y^2=6x^6+x^5+12x^4+12x^2+16x+6$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 178 82948 23402194 6880370704 2013144841618 582622191933700 168358154431328242 48658925715634520064 14063027005371573629746 4064231406644218986637828

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 290 4762 82374 1417850 24137570 410290730 6975432574 118587392074 2015993900450

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{17}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{8})$$.
Endomorphism algebra over $\overline{\F}_{17}$
 The base change of $A$ to $\F_{17^{4}}$ is 1.83521.awc 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
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 the simple isogeny class 2.289.a_awc and its endomorphism algebra is $$\Q(\zeta_{8})$$.

## 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.i_bg $2$ (not in LMFDB) 2.17.am_cs $8$ (not in LMFDB) 2.17.a_ac $8$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.17.i_bg $2$ (not in LMFDB) 2.17.am_cs $8$ (not in LMFDB) 2.17.a_ac $8$ (not in LMFDB) 2.17.a_c $8$ (not in LMFDB) 2.17.m_cs $8$ (not in LMFDB) 2.17.ag_t $24$ (not in LMFDB) 2.17.g_t $24$ (not in LMFDB)