# Properties

 Label 2.5.ac_c 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: $( 1 - 4 x + 5 x^{2} )( 1 + 2 x + 5 x^{2} )$ Frobenius angles: $\pm0.147583617650$, $\pm0.647583617650$ Angle rank: $1$ (numerical) Jacobians: 5

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 5 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 16 640 12688 409600 10272016 244117120 6151673488 153413222400 3812818821136 95367450947200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 26 100 654 3284 15626 78740 392734 1952164 9765626

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae $\times$ 1.5.c 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 $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{4}}$ is 1.625.o 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{5^{2}}$  The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag $\times$ 1.25.g. 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 2.5.ag_s $2$ 2.25.a_o 2.5.c_c $2$ 2.25.a_o 2.5.g_s $2$ 2.25.a_o
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.5.ag_s $2$ 2.25.a_o 2.5.c_c $2$ 2.25.a_o 2.5.g_s $2$ 2.25.a_o 2.5.ai_ba $4$ 2.625.bc_cdq 2.5.ae_o $4$ 2.625.bc_cdq 2.5.a_ag $4$ 2.625.bc_cdq 2.5.a_g $4$ 2.625.bc_cdq 2.5.e_o $4$ 2.625.bc_cdq 2.5.i_ba $4$ 2.625.bc_cdq 2.5.a_ai $8$ (not in LMFDB) 2.5.a_i $8$ (not in LMFDB) 2.5.ae_l $12$ (not in LMFDB) 2.5.ac_ab $12$ (not in LMFDB) 2.5.c_ab $12$ (not in LMFDB) 2.5.e_l $12$ (not in LMFDB)