Properties

Label 2.137.abn_yu
Base field $\F_{137}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{137}$
Dimension:  $2$
L-polynomial:  $1 - 39 x + 644 x^{2} - 5343 x^{3} + 18769 x^{4}$
Frobenius angles:  $\pm0.0784760750967$, $\pm0.254857258237$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{41})\)
Galois group:  $C_2^2$
Jacobians:  $32$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14032$ $347937472$ $6611855780416$ $124103830860589824$ $2329200821682127388752$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $99$ $18537$ $2571354$ $352292113$ $48261864819$ $6611855310222$ $905824268766411$ $124097929543725601$ $17001416405572203978$ $2329194047653348323657$

Jacobians and polarizations

This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{137^{6}}$.

Endomorphism algebra over $\F_{137}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{41})\).
Endomorphism algebra over $\overline{\F}_{137}$
The base change of $A$ to $\F_{137^{6}}$ is 1.6611856250609.abatok 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$
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
2.137.bn_yu$2$(not in LMFDB)
2.137.a_iz$3$(not in LMFDB)
2.137.bn_yu$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.137.bn_yu$2$(not in LMFDB)
2.137.a_iz$3$(not in LMFDB)
2.137.bn_yu$3$(not in LMFDB)
2.137.a_aiz$12$(not in LMFDB)