Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 40 x + 672 x^{2} - 5560 x^{3} + 19321 x^{4}$ |
Frobenius angles: | $\pm0.0989502623149$, $\pm0.232779741831$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.7430400.5 |
Galois group: | $D_{4}$ |
Jacobians: | $32$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14394$ | $368400036$ | $7212443624874$ | $139363117978544400$ | $2692472389058327844954$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $100$ | $19066$ | $2685580$ | $373326358$ | $51889233700$ | $7212552684586$ | $1002544379408140$ | $139353667126146718$ | $19370159742298074820$ | $2692452204247217689306$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=72x^6+111x^5+11x^4+34x^3+132x^2+46x+4$
- $y^2=20x^6+70x^5+24x^4+118x^3+47x^2+79x+25$
- $y^2=102x^6+40x^5+25x^4+26x^3+138x^2+120x+99$
- $y^2=35x^6+133x^5+46x^4+4x^3+53x^2+52x+41$
- $y^2=39x^6+100x^5+18x^4+120x^3+48x^2+109x+56$
- $y^2=10x^6+101x^5+131x^4+43x^3+47x^2+50x+138$
- $y^2=59x^6+59x^5+25x^4+137x^3+119x^2+53x+126$
- $y^2=39x^6+75x^5+65x^4+51x^3+109x+127$
- $y^2=86x^6+64x^5+31x^4+14x^3+10x^2+43x+82$
- $y^2=59x^6+126x^5+52x^4+81x^3+118x^2+50x+102$
- $y^2=9x^6+74x^5+113x^4+110x^3+13x^2+39x+11$
- $y^2=80x^6+93x^5+117x^4+134x^3+100x^2+3x+48$
- $y^2=34x^6+62x^5+132x^4+42x^3+71x^2+7x+49$
- $y^2=35x^6+14x^5+100x^4+32x^3+92x^2+58x+12$
- $y^2=7x^6+22x^5+111x^4+95x^3+117x^2+81x+61$
- $y^2=26x^6+94x^5+125x^4+115x^3+72x^2+68x+97$
- $y^2=48x^6+81x^5+19x^4+54x^3+118x^2+60x+88$
- $y^2=103x^6+28x^5+100x^4+42x^3+37x^2+7x+27$
- $y^2=77x^6+106x^5+108x^4+71x^3+101x^2+106x+60$
- $y^2=95x^6+96x^5+111x^4+15x^3+125x^2+27x+38$
- and 12 more
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 4.0.7430400.5. |
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_zw | $2$ | (not in LMFDB) |