Invariants
Base field: | $\F_{37}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 18 x + 147 x^{2} - 666 x^{3} + 1369 x^{4}$ |
Frobenius angles: | $\pm0.0751079154746$, $\pm0.330643545232$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.56896.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $833$ | $1833433$ | $2571144464$ | $3512184758089$ | $4807609722686393$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $20$ | $1340$ | $50762$ | $1874004$ | $69329900$ | $2565609590$ | $94931587820$ | $3512482272484$ | $129961774501634$ | $4808584550050700$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=32x^6+26x^5+7x^4+28x^3+3x^2+4x+22$
- $y^2=8x^6+22x^5+21x^4+26x^3+21x^2+11x+16$
- $y^2=8x^6+x^5+30x^4+27x^3+7x^2+4x+32$
- $y^2=24x^6+27x^5+19x^4+8x^3+35x^2+16x+35$
- $y^2=19x^6+10x^5+5x^4+22x^3+28x^2+29x+22$
- $y^2=12x^6+5x^5+31x^4+4x^3+29x^2+15x+8$
- $y^2=7x^6+12x^5+36x^3+30x^2+31x+13$
- $y^2=31x^6+3x^5+19x^4+9x^3+7x^2+18x+2$
- $y^2=13x^6+15x^5+16x^4+15x^3+5x^2+12x+17$
- $y^2=20x^6+6x^5+35x^4+15x^3+6x^2+10x+36$
- $y^2=7x^6+12x^5+16x^4+2x^3+6x^2+7x+11$
- $y^2=29x^6+32x^5+34x^4+17x^3+29x^2+20x+29$
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 4.0.56896.1. |
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.s_fr | $2$ | (not in LMFDB) |