Invariants
Base field: | $\F_{139}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 41 x + 694 x^{2} - 5699 x^{3} + 19321 x^{4}$ |
Frobenius angles: | $\pm0.0938786147385$, $\pm0.214217877934$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.733193.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14276$ | $367692656$ | $7210790069072$ | $139361159970884288$ | $2692472987268365181516$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $99$ | $19029$ | $2684964$ | $373321113$ | $51889245229$ | $7212553828086$ | $1002544398533343$ | $139353667276811889$ | $19370159741624168892$ | $2692452204205208789989$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=41x^6+17x^5+111x^4+27x^3+62x^2+116x+43$
- $y^2=83x^6+26x^5+116x^4+68x^3+75x^2+82x+67$
- $y^2=74x^6+25x^5+9x^4+44x^3+105x^2+53x+58$
- $y^2=116x^6+73x^5+57x^4+11x^3+29x^2+132x+40$
- $y^2=x^6+71x^5+77x^4+115x^3+98x^2+62x+22$
- $y^2=57x^6+137x^5+59x^4+115x^3+122x^2+23x+110$
- $y^2=67x^6+128x^5+59x^4+10x^3+131x^2+127x+138$
- $y^2=93x^6+52x^5+87x^4+119x^3+99x^2+93x+9$
- $y^2=64x^6+113x^5+85x^4+75x^3+36x^2+97x+32$
- $y^2=48x^6+90x^5+127x^4+37x^3+96x^2+18x+19$
- $y^2=138x^6+7x^5+46x^4+57x^3+62x^2+133x+17$
- $y^2=17x^6+8x^5+37x^4+112x^3+106x^2+40x+26$
- $y^2=119x^6+79x^5+36x^4+134x^3+54x^2+104x+98$
- $y^2=97x^6+29x^5+18x^4+126x^3+48x^2+54x+85$
- $y^2=64x^6+13x^5+63x^4+91x^3+94x^2+21x+83$
- $y^2=92x^6+9x^5+94x^4+68x^3+50x^2+15x+43$
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 4.0.733193.1. |
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.bp_bas | $2$ | (not in LMFDB) |