# Properties

 Label 2.11.ag_s Base field $\F_{11}$ Dimension $2$ $p$-rank $2$ Ordinary Yes Supersingular No Simple Yes Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{11}$ Dimension: $2$ L-polynomial: $1 - 6 x + 18 x^{2} - 66 x^{3} + 121 x^{4}$ Frobenius angles: $\pm0.0290991158233$, $\pm0.529099115823$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{13})$$ 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:

• $y^2=8x^6+6x^5+x^4+x^2+5x+8$
• $y^2=9x^5+6x^4+3x^3+5x^2+6$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 68 14416 1655732 207821056 25878546548 3138426760144 379509165371012 45940387292319744 5559929409777049892 672749994945773415376

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 122 1242 14190 160686 1771562 19474818 214315294 2357952822 25937424602

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{13})$$.
Endomorphism algebra over $\overline{\F}_{11}$
 The base change of $A$ to $\F_{11^{4}}$ is 1.14641.ais 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-13})$$$)$
All geometric endomorphisms are defined over $\F_{11^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{11^{2}}$  The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 2.121.a_ais and its endomorphism algebra is $$\Q(i, \sqrt{13})$$.

## 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 2.11.g_s $2$ 2.121.a_ais 2.11.a_ae $8$ (not in LMFDB) 2.11.a_e $8$ (not in LMFDB)