Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 51 x + 1032 x^{2} - 9843 x^{3} + 37249 x^{4}$ |
Frobenius angles: | $\pm0.0403859860472$, $\pm0.180450830900$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.49708.1 |
Galois group: | $D_{4}$ |
Jacobians: | $14$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28388$ | $1367620288$ | $51651753544208$ | $1925098953201074944$ | $71708945830468264883588$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $143$ | $36713$ | $7184774$ | $1387470705$ | $267785337143$ | $51682544035454$ | $9974730335278439$ | $1925122951943928865$ | $371548729882682343686$ | $71708904872681924733113$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=160x^6+118x^5+105x^4+38x^3+85x^2+159x+13$
- $y^2=29x^6+60x^5+154x^4+17x^3+68x^2+56x+17$
- $y^2=173x^6+61x^5+73x^4+148x^3+145x^2+47x+59$
- $y^2=28x^6+118x^5+135x^4+23x^3+91x^2+61x+174$
- $y^2=134x^6+16x^5+124x^4+187x^3+16x^2+66x+47$
- $y^2=44x^6+51x^5+160x^4+187x^3+32x^2+188x+156$
- $y^2=146x^6+179x^5+190x^4+102x^3+31x^2+42x+35$
- $y^2=52x^6+91x^5+173x^4+135x^3+149x^2+173x+65$
- $y^2=143x^6+161x^5+30x^4+39x^3+x^2+191x+98$
- $y^2=121x^6+77x^5+97x^4+19x^3+184x^2+135x+186$
- $y^2=170x^6+21x^5+131x^4+132x^3+89x^2+171x+51$
- $y^2=177x^6+58x^5+108x^4+91x^3+146x^2+189x+60$
- $y^2=92x^6+179x^5+167x^4+134x^3+79x^2+166x+68$
- $y^2=80x^6+81x^5+175x^4+153x^3+47x^2+77x+171$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{193}$.
Endomorphism algebra over $\F_{193}$The endomorphism algebra of this simple isogeny class is 4.0.49708.1. |
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 |
---|---|---|
2.193.bz_bns | $2$ | (not in LMFDB) |