Properties

Label 2.11.ai_bg
Base field $\F_{11}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $1 - 8 x + 32 x^{2} - 88 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.0750991438595$, $\pm0.424900856141$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{6})\)
Galois group:  $C_2^2$
Jacobians:  $3$
Isomorphism classes:  4

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $58$ $14500$ $1760938$ $210250000$ $25769191258$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $122$ $1324$ $14358$ $160004$ $1771562$ $19496684$ $214377118$ $2357916004$ $25937424602$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{4}}$.

Endomorphism algebra over $\F_{11}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{6})\).
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{4}}$ is 1.14641.afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
Remainder of endomorphism lattice by field

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.i_bg$2$2.121.a_afm
2.11.i_bg$4$(not in LMFDB)
2.11.a_ak$8$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.11.i_bg$2$2.121.a_afm
2.11.i_bg$4$(not in LMFDB)
2.11.a_ak$8$(not in LMFDB)
2.11.a_k$8$(not in LMFDB)
2.11.ag_x$24$(not in LMFDB)
2.11.g_x$24$(not in LMFDB)