# Properties

 Label 2.5.ag_r 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: $1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4}$ Frobenius angles: $\pm0.0512862249088$, $\pm0.384619558242$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Galois group: $C_2^2$ Jacobians: 1

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 1 curves, and hence is principally polarizable:

• $y^2=2x^6+4x^5+3x^4+x^3+3x^2+3x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 7 553 15484 363321 9198847 239754256 6109689607 152926532073 3814701058588 95315867737993

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 24 126 580 2940 15342 78204 391492 1953126 9760344

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-6})$$$)$
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.ac_av and its endomorphism algebra is $$\Q(\sqrt{2}, \sqrt{-3})$$.
• Endomorphism algebra over $\F_{5^{3}}$  The base change of $A$ to $\F_{5^{3}}$ is the simple isogeny class 2.125.a_afm and its endomorphism algebra is $$\Q(\sqrt{2}, \sqrt{-3})$$.

## 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.a_c $3$ 2.125.a_afm 2.5.g_r $3$ 2.125.a_afm 2.5.a_c $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.a_c $3$ 2.125.a_afm 2.5.g_r $3$ 2.125.a_afm 2.5.a_c $6$ (not in LMFDB) 2.5.a_ac $12$ (not in LMFDB) 2.5.ae_i $24$ (not in LMFDB) 2.5.e_i $24$ (not in LMFDB)