Properties

Label 2.41.aj_ct
Base field $\F_{41}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{41}$
Dimension:  $2$
L-polynomial:  $1 - 9 x + 71 x^{2} - 369 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.211163249537$, $\pm0.527129909542$
Angle rank:  $2$ (numerical)
Number field:  \(\Q(\zeta_{5})\)
Galois group:  $C_4$
Jacobians:  $69$
Isomorphism classes:  91

This isogeny class is simple and 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1375$ $2930125$ $4755796375$ $7984359145125$ $13426146538750000$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $33$ $1743$ $69003$ $2825563$ $115886298$ $4750316583$ $194753838543$ $7984917674803$ $327381925076613$ $13422659245067598$

Jacobians and polarizations

This isogeny class contains the Jacobians of 69 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41}$.

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.41.j_ct$2$(not in LMFDB)
2.41.at_gp$5$(not in LMFDB)
2.41.b_acr$5$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.41.j_ct$2$(not in LMFDB)
2.41.at_gp$5$(not in LMFDB)
2.41.b_acr$5$(not in LMFDB)
2.41.l_eh$5$(not in LMFDB)
2.41.q_ew$5$(not in LMFDB)
2.41.aq_ew$10$(not in LMFDB)
2.41.al_eh$10$(not in LMFDB)
2.41.ab_acr$10$(not in LMFDB)
2.41.t_gp$10$(not in LMFDB)