Invariants
Base field: | $\F_{211}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 50 x + 1032 x^{2} - 10550 x^{3} + 44521 x^{4}$ |
Frobenius angles: | $\pm0.0353230557471$, $\pm0.240811115290$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.302400.2 |
Galois group: | $D_{4}$ |
Jacobians: | $32$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $34954$ | $1962806916$ | $88228556801314$ | $3928813403553771600$ | $174913966291347876337954$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $162$ | $44086$ | $9392082$ | $1982127478$ | $418227139302$ | $88245928095766$ | $18619892917564422$ | $3928797472029141598$ | $828976267859336505042$ | $174913992534797694032806$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=52x^6+62x^5+137x^4+53x^3+163x^2+210x+121$
- $y^2=112x^6+34x^5+146x^4+80x^3+169x^2+88x+10$
- $y^2=145x^6+169x^5+55x^4+147x^3+53x^2+47x+28$
- $y^2=178x^6+17x^5+141x^4+56x^3+49x^2+56x+52$
- $y^2=175x^6+192x^5+4x^4+27x^3+74x^2+66x+18$
- $y^2=89x^6+86x^5+102x^4+183x^3+194x^2+5x+100$
- $y^2=150x^6+186x^5+105x^4+8x^3+195x^2+51x+138$
- $y^2=26x^6+150x^5+113x^4+129x^3+56x^2+15x+84$
- $y^2=175x^6+9x^5+67x^4+89x^3+41x^2+193x+122$
- $y^2=158x^6+42x^5+193x^4+202x^3+173x^2+163x+196$
- $y^2=68x^6+126x^5+88x^4+181x^3+65x^2+19x+134$
- $y^2=26x^6+175x^5+20x^4+193x^3+152x^2+17x+103$
- $y^2=109x^6+122x^5+15x^4+158x^3+167x^2+28x+16$
- $y^2=204x^6+58x^5+16x^4+50x^3+152x^2+203x+150$
- $y^2=202x^6+205x^5+43x^4+187x^3+88x^2+69x+155$
- $y^2=150x^6+148x^5+43x^4+35x^3+38x^2+93x+126$
- $y^2=26x^6+105x^5+180x^4+149x^3+146x^2+157$
- $y^2=193x^6+42x^5+111x^4+152x^3+164x^2+22x+163$
- $y^2=141x^6+67x^5+56x^4+123x^3+116x^2+106x+198$
- $y^2=193x^6+72x^5+182x^4+46x^3+16x^2+30x+149$
- and 12 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{211}$.
Endomorphism algebra over $\F_{211}$The endomorphism algebra of this simple isogeny class is 4.0.302400.2. |
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 |
---|---|---|
2.211.by_bns | $2$ | (not in LMFDB) |