# Properties

 Label 2.137.abo_zw 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: $1 - 40 x + 672 x^{2} - 5480 x^{3} + 18769 x^{4}$ Frobenius angles: $\pm0.132371854234$, $\pm0.208023534643$ Angle rank: $2$ (numerical) Number field: 4.0.1159424.1 Galois group: $D_{4}$ Jacobians: 20

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 20 curves, and hence is principally polarizable:

• $y^2=103x^6+76x^5+66x^4+135x^3+39x^2+42x+21$
• $y^2=70x^6+84x^5+9x^4+41x^3+120x^2+61x+128$
• $y^2=35x^6+53x^5+20x^4+96x^3+70x^2+131x+64$
• $y^2=116x^6+120x^5+132x^4+103x^3+68x^2+39x+116$
• $y^2=83x^6+115x^5+63x^4+133x^3+69x^2+51x+106$
• $y^2=91x^6+8x^5+104x^4+95x^3+60x^2+2x+10$
• $y^2=116x^6+82x^5+28x^4+26x^3+99x^2+123x+45$
• $y^2=61x^6+18x^5+36x^4+123x^3+83x^2+103x+40$
• $y^2=95x^6+112x^5+79x^4+106x^3+80x^2+96x+31$
• $y^2=86x^6+105x^5+18x^4+123x^3+113x^2+26x+93$
• $y^2=35x^6+41x^5+97x^4+40x^3+109x^2+95x+31$
• $y^2=77x^6+11x^5+23x^4+53x^3+10x^2+55x+51$
• $y^2=54x^6+110x^5+41x^4+93x^3+49x^2+15x+16$
• $y^2=112x^6+126x^5+36x^3+92x^2+49x+6$
• $y^2=133x^6+32x^5+115x^4+82x^3+4x^2+14x+117$
• $y^2=92x^6+94x^5+43x^4+45x^3+110x^2+68x+125$
• $y^2=112x^6+39x^5+9x^4+121x^3+57x^2+21x+128$
• $y^2=4x^6+134x^5+58x^4+3x^3+70x^2+11x+58$
• $y^2=113x^6+43x^5+134x^4+103x^3+130x^2+89x+43$
• $y^2=36x^6+28x^5+7x^4+11x^3+126x^2+103x+114$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 13922 347520964 6612374422754 124110580288560656 2329225408221710305282 43716694404522716237378116 820517734511447549344481793218 15400296265074445936170111902941184 289048159782228653018100060016941577826 5425144911105259138097512555758032729161924

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 98 18514 2571554 352311270 48262374258 6611864013298 905824373197394 124097930312659134 17001416404782209378 2329194047520683008914

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{137}$
 The endomorphism algebra of this simple isogeny class is 4.0.1159424.1.
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.bo_zw $2$ (not in LMFDB)