# Properties

 Label 2.137.abp_bak Base Field $\F_{137}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{137}$ Dimension: $2$ L-polynomial: $1 - 41 x + 686 x^{2} - 5617 x^{3} + 18769 x^{4}$ Frobenius angles: $\pm0.0179276178433$, $\pm0.228596896516$ Angle rank: $2$ (numerical) Number field: 4.0.448668.1 Galois group: $D_{4}$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2=54x^6+53x^5+65x^4+79x^3+126x^2+33x+109$
• $y^2=48x^6+23x^5+24x^4+51x^3+20x+127$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 13798 346522972 6608271338296 124097788085696704 2329192779126488895478 43716624347913816886705408 820517606724080334269317926502 15400296068470250290976354969555712 289048159535154087608804116889119628152 5425144910874606289843272163751621845815772

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 97 18461 2569960 352274961 48261698177 6611853417698 905824232124377 124097928728392609 17001416390249622232 2329194047421656116541

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{137}$
 The endomorphism algebra of this simple isogeny class is 4.0.448668.1.
All geometric endomorphisms are defined over $\F_{137}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.137.bp_bak $2$ (not in LMFDB)