Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 40 x + 675 x^{2} - 5560 x^{3} + 19321 x^{4}$ |
Frobenius angles: | $\pm0.126846357897$, $\pm0.217883959392$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2679696.1 |
Galois group: | $D_{4}$ |
Jacobians: | $18$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14397$ | $368520009$ | $7213412355012$ | $139367268840309081$ | $2692484593399204872477$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $100$ | $19072$ | $2685940$ | $373337476$ | $51889468900$ | $7212556397662$ | $1002544422864700$ | $139353667441169668$ | $19370159741633281900$ | $2692452204178120224352$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=130x^6+78x^5+124x^4+27x^3+88x^2+90x+76$
- $y^2=49x^6+36x^5+83x^4+92x^3+130x^2+21x+79$
- $y^2=125x^6+24x^5+42x^4+127x^3+137x^2+66x+10$
- $y^2=3x^6+96x^5+93x^4+16x^2+112x+31$
- $y^2=108x^6+55x^5+129x^4+26x^3+61x^2+102x+68$
- $y^2=111x^6+110x^5+83x^4+63x^3+6x^2+84x+28$
- $y^2=56x^6+94x^5+6x^4+44x^3+2x^2+61x+21$
- $y^2=136x^6+64x^5+80x^4+123x^2+56x+27$
- $y^2=15x^6+86x^5+18x^4+46x^3+5x^2+81x+130$
- $y^2=74x^6+50x^5+78x^4+119x^3+70x^2+106x+128$
- $y^2=59x^6+4x^5+26x^4+23x^3+108x^2+65x+133$
- $y^2=59x^6+123x^5+89x^4+73x^3+128x^2+15x+27$
- $y^2=134x^6+43x^5+5x^4+112x^3+134x^2+20x+44$
- $y^2=68x^6+133x^5+32x^4+96x^3+84x^2+133x+115$
- $y^2=43x^6+50x^5+27x^4+90x^3+72x^2+138x+105$
- $y^2=60x^6+125x^5+38x^4+35x^3+79x^2+50x+91$
- $y^2=14x^6+20x^5+99x^4+79x^3+64x^2+98x+88$
- $y^2=21x^6+97x^5+16x^4+15x^3+111x^2+72x+39$
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 4.0.2679696.1. |
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_zz | $2$ | (not in LMFDB) |