Properties

Label 2.137.abn_yu
Base Field $\F_{137}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
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.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 32 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 14032 347937472 6611855780416 124103830860589824 2329200821682127388752 43716636861020472409133056 820517639915323894020391512784 15400296169651386256229847210470400 289048159795659680197698116558117099584 5425144911414262399520881728503982797225152

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

Decomposition and endomorphism algebra

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 $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$
All geometric endomorphisms are defined over $\F_{137^{6}}$.
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
2.137.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)