Invariants
Base field: | $\F_{113}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 38 x + 585 x^{2} - 4294 x^{3} + 12769 x^{4}$ |
Frobenius angles: | $\pm0.0901024562033$, $\pm0.189951722793$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.322112.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9023$ | $159589801$ | $2080418564924$ | $26585707627025561$ | $339460904063616474223$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $76$ | $12496$ | $1441834$ | $163055124$ | $18424577856$ | $2081954732086$ | $235260575934880$ | $26584442117490468$ | $3004041938736059818$ | $339456738992276095296$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=28x^6+27x^5+111x^4+45x^3+17x^2+76x+70$
- $y^2=93x^6+20x^5+45x^4+82x^3+31x^2+79x+33$
- $y^2=8x^6+43x^5+11x^4+77x^3+34x^2+7x+20$
- $y^2=102x^6+53x^5+31x^4+52x^3+69x^2+6x+56$
- $y^2=66x^6+21x^5+19x^4+10x^3+4x^2+89x+59$
- $y^2=12x^6+19x^5+5x^4+60x^3+31x^2+73x+57$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{113}$.
Endomorphism algebra over $\F_{113}$The endomorphism algebra of this simple isogeny class is 4.0.322112.1. |
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 |
---|---|---|
2.113.bm_wn | $2$ | (not in LMFDB) |