Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 49 x + 994 x^{2} - 9751 x^{3} + 39601 x^{4}$ |
Frobenius angles: | $\pm0.109464524120$, $\pm0.207308250412$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1912313.2 |
Galois group: | $D_{4}$ |
Jacobians: | $24$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30796$ | $1551995216$ | $62097671921392$ | $2459456857100257088$ | $97394075393379673594516$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $151$ | $39189$ | $7879816$ | $1568291913$ | $312080876421$ | $62103859410126$ | $12358664478179947$ | $2459374192955793009$ | $489415464122898455128$ | $97393677359684032621789$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=91x^6+63x^5+55x^4+92x^3+33x^2+34x+136$
- $y^2=10x^6+41x^5+119x^4+168x^3+72x^2+192x+9$
- $y^2=119x^6+104x^5+191x^4+111x^3+51x^2+146x+173$
- $y^2=114x^6+160x^5+111x^4+2x^3+14x^2+8x+141$
- $y^2=156x^6+46x^5+61x^4+54x^3+94x^2+163x+90$
- $y^2=39x^6+x^5+137x^4+196x^3+27x^2+65x+2$
- $y^2=20x^6+6x^5+141x^4+94x^3+119x^2+175x+54$
- $y^2=140x^6+21x^5+144x^4+36x^3+76x^2+108x+87$
- $y^2=151x^6+85x^5+36x^4+34x^3+174x^2+92x+76$
- $y^2=146x^6+119x^5+68x^4+120x^3+178x^2+92x+146$
- $y^2=75x^6+116x^5+190x^4+108x^3+5x^2+3x+80$
- $y^2=94x^6+149x^5+130x^4+37x^3+180x^2+31x+145$
- $y^2=99x^6+193x^5+105x^4+182x^3+158x^2+55x$
- $y^2=172x^6+81x^5+22x^4+36x^3+98x^2+143x+62$
- $y^2=68x^6+135x^5+93x^4+177x^3+190x^2+46x+137$
- $y^2=126x^6+173x^5+75x^4+169x^3+84x^2+102x+175$
- $y^2=45x^6+81x^5+146x^4+164x^3+153x^2+99x+58$
- $y^2=105x^6+76x^5+30x^4+150x^3+82x^2+138x+134$
- $y^2=x^6+167x^5+157x^4+194x^3+142x^2+135x+73$
- $y^2=29x^6+101x^5+198x^4+149x^3+73x^2+98x+90$
- $y^2=38x^6+47x^5+184x^4+79x^3+50x^2+123x+76$
- $y^2=51x^6+191x^5+164x^4+53x^3+112x^2+152x+165$
- $y^2=49x^6+5x^5+59x^4+191x^3+24x^2+41x+16$
- $y^2=164x^6+174x^5+183x^4+175x^3+56x^2+105x+193$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{199}$.
Endomorphism algebra over $\F_{199}$The endomorphism algebra of this simple isogeny class is 4.0.1912313.2. |
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 |
---|---|---|
2.199.bx_bmg | $2$ | (not in LMFDB) |