Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 40 x + 666 x^{2} - 5560 x^{3} + 19321 x^{4}$ |
Frobenius angles: | $\pm0.0315253631714$, $\pm0.252613043445$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.13824.1 |
Galois group: | $D_{4}$ |
Jacobians: | $30$ |
This isogeny class is simple and 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14388$ | $368160144$ | $7210506273300$ | $139354776032169984$ | $2692447420025510720628$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $100$ | $19054$ | $2684860$ | $373304014$ | $51888752500$ | $7212544698238$ | $1002544272445900$ | $139353665942429854$ | $19370159731145294980$ | $2692452204148542880654$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 30 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=14x^6+61x^5+68x^4+38x^3+41x^2+138x+67$
- $y^2=89x^6+137x^5+78x^4+97x^3+101x^2+72$
- $y^2=66x^6+81x^5+129x^4+48x^3+115x^2+90x+2$
- $y^2=59x^6+21x^5+29x^4+50x^3+92x^2+51x+32$
- $y^2=77x^6+32x^5+101x^4+129x^3+33x^2+26x+21$
- $y^2=106x^6+77x^5+58x^4+112x^3+80x^2+33x+108$
- $y^2=36x^6+74x^5+65x^4+55x^3+84x^2+36x+88$
- $y^2=92x^6+95x^5+26x^4+52x^3+102x^2+105x+106$
- $y^2=23x^6+23x^5+121x^4+18x^3+121x^2+120x+130$
- $y^2=109x^6+114x^5+54x^4+30x^3+41x^2+109x+73$
- $y^2=43x^6+81x^5+112x^4+32x^3+124x^2+80x+61$
- $y^2=111x^6+76x^5+20x^4+29x^3+123x^2+26x+121$
- $y^2=131x^6+115x^5+90x^4+94x^3+82x^2+13x+43$
- $y^2=64x^6+117x^5+110x^4+49x^3+61x^2+2x+86$
- $y^2=19x^6+12x^5+109x^4+126x^3+111x^2+55x+28$
- $y^2=80x^6+51x^5+59x^4+138x^3+105x^2+133x+39$
- $y^2=20x^6+22x^5+119x^4+109x^3+60x^2+62x+98$
- $y^2=103x^6+100x^5+6x^4+36x^3+63x^2+95x+77$
- $y^2=114x^6+131x^5+88x^4+130x^3+45x^2+67x+12$
- $y^2=68x^6+37x^5+94x^4+82x^3+88x^2+10$
- $y^2=71x^6+94x^5+30x^4+x^3+84x^2+126x+113$
- $y^2=33x^6+103x^5+86x^4+88x^3+105x^2+86x+61$
- $y^2=109x^6+74x^5+23x^4+56x^3+15x^2+113x+47$
- $y^2=56x^6+112x^5+10x^4+21x^3+35x^2+34x+31$
- $y^2=76x^6+18x^5+65x^4+74x^3+54x^2+29x+109$
- $y^2=71x^6+62x^5+83x^4+30x^3+47x^2+18x+19$
- $y^2=135x^6+100x^5+136x^4+102x^3+121x^2+82x+33$
- $y^2=30x^6+3x^5+114x^4+87x^3+75x^2+128x+17$
- $y^2=99x^6+86x^5+75x^4+135x^3+122x^2+137x+50$
- $y^2=34x^6+100x^5+94x^4+101x^3+60x^2+20x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{139}$.
Endomorphism algebra over $\F_{139}$The endomorphism algebra of this simple isogeny class is 4.0.13824.1. |
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 |
---|---|---|
2.139.bo_zq | $2$ | (not in LMFDB) |