Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 40 x + 671 x^{2} - 5560 x^{3} + 19321 x^{4}$ |
Frobenius angles: | $\pm0.0898857305018$, $\pm0.236719804447$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.8624784.2 |
Galois group: | $D_{4}$ |
Jacobians: | $16$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14393$ | $368360049$ | $7212120722900$ | $139361731378539849$ | $2692468279437281892353$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $100$ | $19064$ | $2685460$ | $373322644$ | $51889154500$ | $7212551405438$ | $1002544363443100$ | $139353666980531044$ | $19370159741615128780$ | $2692452204253471224104$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=117x^6+27x^5+104x^4+120x^3+50x^2+121x+67$
- $y^2=133x^6+74x^5+9x^4+25x^3+115x^2+48x+32$
- $y^2=136x^6+11x^5+58x^4+39x^3+x^2+84x+40$
- $y^2=60x^6+88x^5+66x^4+22x^3+112x^2+93x+18$
- $y^2=73x^6+80x^5+27x^4+34x^3+24x^2+44x+49$
- $y^2=59x^6+78x^5+37x^4+69x^3+99x^2+111x+100$
- $y^2=23x^6+91x^5+133x^4+112x^3+117x^2+70x+53$
- $y^2=80x^6+111x^5+65x^4+50x^3+25x^2+138x+36$
- $y^2=25x^6+25x^5+6x^4+51x^3+13x^2+68x+98$
- $y^2=92x^6+31x^5+33x^4+92x^3+65x^2+51x+69$
- $y^2=96x^6+3x^5+128x^4+114x^3+41x^2+42x+135$
- $y^2=32x^6+98x^5+68x^4+76x^3+50x^2+128x+27$
- $y^2=61x^6+127x^5+36x^4+123x^3+54x^2+23x+60$
- $y^2=72x^6+61x^5+39x^4+x^3+91x^2+45x+38$
- $y^2=72x^6+57x^5+105x^3+36x^2+108x+32$
- $y^2=132x^6+48x^5+27x^4+126x^3+94x^2+23x+59$
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 4.0.8624784.2. |
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_zv | $2$ | (not in LMFDB) |