Invariants
Base field: | $\F_{37}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 17 x + 142 x^{2} - 629 x^{3} + 1369 x^{4}$ |
Frobenius angles: | $\pm0.165306334649$, $\pm0.322478204771$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1122476.1 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
Isomorphism classes: | 10 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $866$ | $1868828$ | $2588214200$ | $3518121037184$ | $4809172673879946$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $21$ | $1365$ | $51096$ | $1877169$ | $69352441$ | $2565728490$ | $94931994213$ | $3512482368129$ | $129961762406472$ | $4808584413789325$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=13x^6+23x^5+33x^4+x^3+19x^2+27x+9$
- $y^2=33x^6+5x^5+9x^4+14x^3+14x^2+25x+5$
- $y^2=11x^6+24x^5+2x^4+10x^3+35x^2+14x+18$
- $y^2=35x^6+5x^5+12x^4+x^3+2x^2+26x+22$
- $y^2=29x^6+8x^5+33x^4+19x^3+17x^2+20x+25$
- $y^2=20x^6+20x^5+9x^4+24x^3+2x^2+10x+25$
- $y^2=22x^6+16x^5+2x^4+8x^3+15x^2+11x+20$
- $y^2=8x^6+26x^5+28x^4+31x^3+x^2+6x+31$
- $y^2=18x^6+30x^5+32x^4+24x^3+13x^2+7x+24$
- $y^2=24x^6+x^5+19x^4+29x^3+20x^2+27x+4$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37}$.
Endomorphism algebra over $\F_{37}$The endomorphism algebra of this simple isogeny class is 4.0.1122476.1. |
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 |
---|---|---|
2.37.r_fm | $2$ | (not in LMFDB) |