# Properties

 Label 2.5.ad_i Base Field $\F_{5}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{5}$ Dimension: $2$ L-polynomial: $1 - 3 x + 8 x^{2} - 15 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.206741677780$, $\pm0.540075011113$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Galois group: $C_2^2$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 16 832 15808 389376 10332496 249892864 6077147344 152101168128 3814694762944 95326492186432

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 33 126 625 3303 15990 77787 389377 1953126 9761433

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{17})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ha 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-51})$$$)$
All geometric endomorphisms are defined over $\F_{5^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{5^{2}}$  The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 2.25.h_y and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{17})$$.
• Endomorphism algebra over $\F_{5^{3}}$  The base change of $A$ to $\F_{5^{3}}$ is the simple isogeny class 2.125.a_ha and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{17})$$.

## 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 2.5.d_i $2$ 2.25.h_y 2.5.a_ah $3$ 2.125.a_ha 2.5.d_i $3$ 2.125.a_ha
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.d_i $2$ 2.25.h_y 2.5.a_ah $3$ 2.125.a_ha 2.5.d_i $3$ 2.125.a_ha 2.5.a_h $12$ (not in LMFDB)