# Properties

 Label 2.11.aj_bp 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: $1 - 9 x + 41 x^{2} - 99 x^{3} + 121 x^{4}$ Frobenius angles: $\pm0.178435994483$, $\pm0.329700688269$ Angle rank: $2$ (numerical) Number field: $$\Q(\zeta_{5})$$ Galois group: $C_4$ Jacobians: 3

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 3 curves, and hence is principally polarizable:

• $y^2=10x^5+9$
• $y^2=6x^6+9x^5+9x^4+8x^2+3x+6$
• $y^2=7x^6+2x^5+3x^4+3x^2+x+8$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 55 14905 1882705 218522205 26001250000 3138320501305 379774219636405 45953777683469205 5560070301147558955 672748459337020000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 123 1413 14923 161448 1771503 19488423 214377763 2358012573 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.j_bp $2$ 2.121.b_fl 2.11.ae_g $5$ (not in LMFDB) 2.11.b_aj $5$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.11.j_bp $2$ 2.121.b_fl 2.11.ae_g $5$ (not in LMFDB) 2.11.b_aj $5$ (not in LMFDB) 2.11.b_v $5$ (not in LMFDB) 2.11.l_bz $5$ (not in LMFDB) 2.11.al_bz $10$ (not in LMFDB) 2.11.ab_aj $10$ (not in LMFDB) 2.11.ab_v $10$ (not in LMFDB) 2.11.e_g $10$ (not in LMFDB)