Properties

Label 2.11.al_bz
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:  $1 - 11 x + 51 x^{2} - 121 x^{3} + 121 x^{4}$
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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)