# Properties

 Label 2.11.al_bz 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: $1 - 11 x + 51 x^{2} - 121 x^{3} + 121 x^{4}$ Frobenius angles: $\pm0.0215640055172$, $\pm0.270299311731$ Angle rank: $2$ (numerical) Number field: $$\Q(\zeta_{5})$$ Galois group: $C_4$ Jacobians: 1

This isogeny class is simple and 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:

• $y^2=8x^5+10$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 41 12505 1755251 214348205 25873696816 3132341948305 379434616027991 45938876013886805 5559681522703812821 672748459337020000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 1 103 1321 14643 160656 1768123 19470991 214308243 2357847691 25937365398

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{5})$$.
All geometric endomorphisms are defined over $\F_{11}$.

## 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.l_bz $2$ 2.121.at_gz 2.11.ab_aj $5$ (not in LMFDB) 2.11.ab_v $5$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.11.l_bz $2$ 2.121.at_gz 2.11.ab_aj $5$ (not in LMFDB) 2.11.ab_v $5$ (not in LMFDB) 2.11.e_g $5$ (not in LMFDB) 2.11.j_bp $5$ (not in LMFDB) 2.11.aj_bp $10$ (not in LMFDB) 2.11.ae_g $10$ (not in LMFDB) 2.11.b_aj $10$ (not in LMFDB) 2.11.b_v $10$ (not in LMFDB)