# Properties

 Label 2.11.aj_bm Base Field $\F_{11}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{11}$ Dimension: $2$ L-polynomial: $1 - 9 x + 38 x^{2} - 99 x^{3} + 121 x^{4}$ Frobenius angles: $\pm0.0468428922585$, $\pm0.380176225592$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{17})$$ 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:

• $y^2=7x^6+8x^5+9x^4+4x^3+8x^2+x+7$
• $y^2=8x^6+2x^5+8x^4+2x^2+3x+5$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 52 13936 1769872 211214016 25719331612 3132446896384 379744766941228 45953547054914304 5559917317647236752 672742381710223208176

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 117 1332 14425 159693 1768182 19486911 214376689 2357947692 25937131077

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{17})$$.
Endomorphism algebra over $\overline{\F}_{11}$
 The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.acna 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-51})$$$)$
All geometric endomorphisms are defined over $\F_{11^{6}}$.
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.af_ads and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{17})$$.
• Endomorphism algebra over $\F_{11^{3}}$  The base change of $A$ to $\F_{11^{3}}$ is the simple isogeny class 2.1331.a_acna 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 2.11.a_f $3$ (not in LMFDB) 2.11.j_bm $3$ (not in LMFDB) 2.11.a_f $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.11.a_f $3$ (not in LMFDB) 2.11.j_bm $3$ (not in LMFDB) 2.11.a_f $6$ (not in LMFDB) 2.11.a_af $12$ (not in LMFDB)