Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 48 x + 949 x^{2} - 9264 x^{3} + 37249 x^{4}$ |
Frobenius angles: | $\pm0.0361850571721$, $\pm0.237645444107$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.66417.2 |
Galois group: | $D_{4}$ |
Jacobians: | $40$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28887$ | $1372450257$ | $51670115428464$ | $1925132210862845049$ | $71708901389715393421647$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $146$ | $36844$ | $7187330$ | $1387494676$ | $267785171186$ | $51682532706142$ | $9974730088074674$ | $1925122948572321124$ | $371548729858344850946$ | $71708904872851032565084$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 40 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=186x^6+87x^5+107x^4+60x^3+99x^2+135x+142$
- $y^2=94x^6+31x^5+176x^4+26x^3+69x^2+92x+78$
- $y^2=31x^6+133x^5+175x^4+112x^3+115x^2+115x+25$
- $y^2=177x^6+155x^5+151x^4+53x^3+51x^2+x+171$
- $y^2=83x^6+28x^5+111x^4+114x^3+48x^2+60x+24$
- $y^2=101x^6+15x^5+173x^4+x^3+160x^2+120x+16$
- $y^2=12x^6+181x^5+57x^4+13x^3+2x^2+146x+78$
- $y^2=30x^6+146x^5+11x^4+107x^3+141x^2+183x+6$
- $y^2=189x^6+170x^5+7x^4+114x^3+150x^2+121x+69$
- $y^2=179x^6+64x^5+192x^4+7x^3+54x^2+139x+63$
- $y^2=107x^6+189x^5+31x^4+13x^3+97x^2+161x+117$
- $y^2=190x^6+42x^5+74x^4+153x^3+174x^2+92x+73$
- $y^2=x^6+120x^5+8x^4+12x^3+17x^2+123x+142$
- $y^2=22x^6+12x^5+166x^4+78x^3+34x^2+87x+130$
- $y^2=154x^6+83x^5+173x^4+63x^3+83x^2+101x+66$
- $y^2=109x^6+85x^5+147x^4+115x^3+148x^2+114x+155$
- $y^2=173x^6+104x^5+93x^4+37x^3+133x^2+137x+127$
- $y^2=26x^6+86x^5+178x^4+128x^3+8x^2+57x+146$
- $y^2=66x^6+149x^5+46x^4+41x^3+163x^2+x+35$
- $y^2=51x^6+65x^5+187x^4+185x^3+171x^2+66x+48$
- and 20 more
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 4.0.66417.2. |
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 |
---|---|---|
2.193.bw_bkn | $2$ | (not in LMFDB) |