Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 47 x + 889 x^{2} - 7943 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.104248499198$, $\pm0.169935694661$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.306125.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21461$ | $803521301$ | $23286978302621$ | $665429532354810245$ | $19005061768865309085696$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $123$ | $28131$ | $4824507$ | $815746563$ | $137859202678$ | $23298098465811$ | $3937376572510707$ | $665416611294768963$ | $112455406970810835843$ | $19004963774992452783406$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+12)x^6+(3a+1)x^5+(9a+11)x^4+(7a+8)x^3+(3a+9)x^2+(a+5)x+4a+8$
- $y^2=(3a+10)x^6+(3a+6)x^5+(8a+2)x^4+(12a+12)x^3+(11a+3)x^2+(10a+2)x+8a+5$
- $y^2=(11a+2)x^6+(4a+11)x^5+(11a+5)x^4+(6a+8)x^3+10ax^2+(9a+8)x+5a+6$
- $y^2=(4a+5)x^6+(12a+3)x^5+6ax^4+(4a+1)x^3+(10a+12)x^2+(5a+2)x+9$
- $y^2=(12a+2)x^6+11x^5+12ax^4+(11a+6)x^3+(8a+8)x^2+(a+9)x+8a+1$
- $y^2=(12a+10)x^6+(5a+1)x^5+(3a+3)x^4+(4a+11)x^3+(5a+6)x^2+(4a+2)x+a+12$
- $y^2=9ax^6+(5a+4)x^5+2x^4+(7a+10)x^3+7ax^2+(6a+5)x+10a+6$
- $y^2=(a+11)x^6+(a+2)x^5+(3a+11)x^4+(a+8)x^3+(3a+4)x^2+(11a+2)x+5a+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.306125.1. |
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 |
---|---|---|
2.169.bv_bif | $2$ | (not in LMFDB) |