Properties

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

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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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)