Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 27 x + 193 x^{2} )( 1 - 22 x + 193 x^{2} )$ |
$1 - 49 x + 980 x^{2} - 9457 x^{3} + 37249 x^{4}$ | |
Frobenius angles: | $\pm0.0758389534121$, $\pm0.209145594264$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $27$ |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28724$ | $1371168864$ | $51668455909184$ | $1925153121045338496$ | $71709076489180543513364$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $145$ | $36809$ | $7187098$ | $1387509745$ | $267785825065$ | $51682548534878$ | $9974730367878841$ | $1925122952395687969$ | $371548729898333227354$ | $71708904873153875140889$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 27 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=120x^6+23x^5+24x^4+138x^3+171x^2+103x+137$
- $y^2=75x^6+58x^5+93x^4+169x^3+63x^2+90x+20$
- $y^2=79x^6+49x^5+77x^4+19x^3+98x^2+97x+119$
- $y^2=108x^6+148x^5+117x^4+44x^3+125x^2+178x+1$
- $y^2=66x^6+147x^5+187x^4+82x^3+191x^2+186x+87$
- $y^2=22x^6+192x^5+35x^4+153x^3+167x^2+32x+45$
- $y^2=30x^6+158x^5+47x^4+7x^3+152x^2+140x+7$
- $y^2=156x^6+101x^5+87x^4+49x^3+97x^2+190x+132$
- $y^2=116x^6+138x^5+41x^4+64x^3+27x^2+169x+31$
- $y^2=60x^6+125x^5+163x^4+12x^3+133x^2+58x+184$
- $y^2=118x^6+155x^5+145x^4+33x^3+17x^2+99x+186$
- $y^2=157x^6+31x^5+134x^4+114x^3+26x^2+155x+44$
- $y^2=173x^6+172x^5+113x^4+184x^3+134x^2+4x+60$
- $y^2=119x^6+127x^5+12x^4+139x^3+56x^2+186x+174$
- $y^2=175x^6+184x^5+54x^4+158x^3+135x^2+90x+52$
- $y^2=55x^6+75x^5+63x^4+126x^3+25x^2+99x+4$
- $y^2=105x^6+48x^5+57x^4+176x^3+63x^2+93x+41$
- $y^2=161x^6+30x^5+26x^4+68x^3+90x^2+159x+149$
- $y^2=43x^6+80x^5+86x^4+77x^3+165x^2+101x+55$
- $y^2=114x^6+65x^5+34x^4+7x^3+50x^2+54x+91$
- $y^2=169x^6+72x^5+54x^4+77x^3+116x^2+36x+44$
- $y^2=21x^6+31x^5+65x^4+77x^3+51x^2+164x+28$
- $y^2=155x^6+65x^5+87x^4+157x^3+131x^2+61x+123$
- $y^2=57x^6+75x^5+26x^4+28x^3+52x^2+104x+30$
- $y^2=153x^6+45x^5+49x^3+9x^2+45x+175$
- $y^2=15x^6+28x^5+53x^4+104x^3+48x^2+107x+32$
- $y^2=119x^6+33x^5+146x^4+87x^3+16x^2+181x+170$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{193}$.
Endomorphism algebra over $\F_{193}$The isogeny class factors as 1.193.abb $\times$ 1.193.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.af_aia | $2$ | (not in LMFDB) |
2.193.f_aia | $2$ | (not in LMFDB) |
2.193.bx_bls | $2$ | (not in LMFDB) |