# Properties

 Label 2.13.ai_bg Base Field $\F_{13}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{13}$ Dimension: $2$ L-polynomial: $1 - 8 x + 32 x^{2} - 104 x^{3} + 169 x^{4}$ Frobenius angles: $\pm0.0370621216586$, $\pm0.462937878341$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{10})$$ Galois group: $C_2^2$ Jacobians: 3

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 3 curves, and hence is principally polarizable:

• $y^2=11x^6+x^5+10x^4+2x^3+6x^2+11x+6$
• $y^2=5x^6+5x^5+6x^4+2x^2+10x+6$
• $y^2=6x^6+12x^5+4x^4+4x^2+x+6$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 90 28260 4705290 798627600 137233172250 23298083455140 3937418050935690 665361008507289600 112452424073098948890 19004963775070323766500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 170 2142 27958 369606 4826810 62749182 815662558 10604218086 137858491850

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{10})$$.
Endomorphism algebra over $\overline{\F}_{13}$
 The base change of $A$ to $\F_{13^{4}}$ is 1.28561.alq 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-10})$$$)$
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{13^{2}}$  The base change of $A$ to $\F_{13^{2}}$ is the simple isogeny class 2.169.a_alq and its endomorphism algebra is $$\Q(i, \sqrt{10})$$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.13.i_bg $2$ 2.169.a_alq 2.13.a_ag $8$ (not in LMFDB) 2.13.a_g $8$ (not in LMFDB)