# Properties

 Label 2.137.abp_bao 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: $1 - 41 x + 690 x^{2} - 5617 x^{3} + 18769 x^{4}$ Frobenius angles: $\pm0.0859306910897$, $\pm0.211296820890$ Angle rank: $2$ (numerical) Number field: 4.0.2343212.1 Galois group: $D_{4}$ Jacobians: 10

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 10 curves, and hence is principally polarizable:

• $y^2=105x^6+117x^5+87x^4+65x^3+135x^2+58$
• $y^2=122x^6+130x^5+113x^4+30x^3+2x^2+40x+89$
• $y^2=67x^6+33x^5+43x^4+63x^3+99x^2+68x+49$
• $y^2=79x^6+58x^5+120x^4+26x^3+4x^2+136x+99$
• $y^2=67x^6+56x^5+65x^4+4x^3+47x^2+103x+116$
• $y^2=x^6+41x^5+37x^4+115x^3+69x^2+43x+64$
• $y^2=115x^6+83x^5+126x^4+52x^3+81x^2+37x+75$
• $y^2=121x^6+125x^5+28x^4+93x^3+133x^2+5x+6$
• $y^2=79x^6+84x^5+63x^4+51x^3+12x^2+12x+131$
• $y^2=86x^6+34x^5+116x^4+37x^3+99x^2+132x+114$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 13802 346678636 6609539135816 124103385773447936 2329210271231295013402 43716667406553962868800512 820517694621509077415764409306 15400296221596321646220073271241728 289048159768238348259926978539351199432 5425144911197141292293425243443830592476716

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 97 18469 2570452 352290849 48262060617 6611859930034 905824329160225 124097929962305793 17001416403959318836 2329194047560131055029

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{137}$
 The endomorphism algebra of this simple isogeny class is 4.0.2343212.1.
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_bao $2$ (not in LMFDB)