# Properties

 Label 2.17.am_cn 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: $1 - 12 x + 65 x^{2} - 204 x^{3} + 289 x^{4}$ Frobenius angles: $\pm0.0157896134134$, $\pm0.349122946747$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{5})$$ 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 139 79369 24128176 6943914441 2010360903259 582168877086976 168359278701453979 48661044555091420809 14063084451918975425584 4064226529530741241632649

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 276 4914 83140 1415886 24118782 410293470 6975736324 118587876498 2015991481236

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{17}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{17}$
 The base change of $A$ to $\F_{17^{6}}$ is 1.24137569.anxi 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
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.ao_adp and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{17^{3}}$  The base change of $A$ to $\F_{17^{3}}$ is the simple isogeny class 2.4913.a_anxi and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{5})$$.

## 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.m_cn $2$ (not in LMFDB) 2.17.a_o $3$ (not in LMFDB) 2.17.m_cn $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.17.m_cn $2$ (not in LMFDB) 2.17.a_o $3$ (not in LMFDB) 2.17.m_cn $3$ (not in LMFDB) 2.17.a_o $6$ (not in LMFDB) 2.17.a_ao $12$ (not in LMFDB)