Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 50 x + 1005 x^{2} - 9650 x^{3} + 37249 x^{4}$ |
Frobenius angles: | $\pm0.0495084158571$, $\pm0.198591147249$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2811456.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28555$ | $1369355025$ | $51659542330540$ | $1925121489680375625$ | $71708984472197459180275$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $144$ | $36760$ | $7185858$ | $1387486948$ | $267785481444$ | $51682543843030$ | $9974730300983508$ | $1925122951265372548$ | $371548729876541564994$ | $71708904872735620403800$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=177x^6+59x^5+52x^4+81x^3+21x^2+124x+24$
- $y^2=14x^6+139x^5+95x^4+185x^3+16x^2+103x+76$
- $y^2=26x^6+147x^5+147x^4+112x^3+5x^2+156x+76$
- $y^2=40x^6+127x^5+50x^4+65x^3+18x^2+119x+88$
- $y^2=160x^6+30x^5+83x^4+169x^3+146x^2+166x+39$
- $y^2=4x^6+51x^5+111x^4+36x^3+61x^2+68x+11$
- $y^2=57x^6+154x^5+76x^4+134x^3+165x^2+92x+171$
- $y^2=38x^6+171x^5+3x^4+106x^3+18x^2+9x+8$
- $y^2=174x^6+89x^5+180x^4+85x^3+181x^2+10x+182$
- $y^2=69x^6+178x^5+23x^4+173x^3+144x^2+148x+154$
- $y^2=159x^6+53x^5+11x^4+114x^3+61x^2+72x+127$
- $y^2=182x^6+7x^5+93x^4+123x^3+79x^2+176x+25$
- $y^2=142x^6+133x^5+31x^4+47x^3+67x^2+98x+119$
- $y^2=13x^6+102x^5+113x^4+82x^3+139x^2+106x+159$
- $y^2=22x^6+77x^5+101x^4+126x^3+187x^2+79x+162$
- $y^2=114x^6+17x^5+187x^4+7x^3+19x^2+42x+106$
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 4.0.2811456.1. |
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.by_bmr | $2$ | (not in LMFDB) |