Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 45 x + 841 x^{2} - 7605 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.115648327740$, $\pm0.206920732394$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.4901.1 |
Galois group: | $D_{4}$ |
Jacobians: | $45$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21753$ | $806013909$ | $23296136610825$ | $665451080819896869$ | $19005090453458068800528$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $125$ | $28219$ | $4826405$ | $815772979$ | $137859410750$ | $23298097651243$ | $3937376509630325$ | $665416610004720739$ | $112455406953737257565$ | $19004963774854639022254$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 45 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+2)x^6+(7a+5)x^5+(10a+9)x^4+(7a+12)x^3+(5a+8)x^2+(2a+8)x+4a+2$
- $y^2=(4a+11)x^6+(8a+11)x^5+(2a+2)x^4+(12a+12)x^3+(7a+11)x^2+5x+10a+8$
- $y^2=(7a+2)x^6+(6a+9)x^5+(7a+3)x^4+(3a+5)x^3+6ax^2+(12a+3)x+5a+8$
- $y^2=(a+3)x^6+(2a+2)x^5+(3a+9)x^4+(11a+11)x^3+(2a+6)x^2+8x+8a+7$
- $y^2=(7a+7)x^6+(6a+7)x^5+(9a+4)x^4+(3a+10)x^3+8x^2+(4a+7)x+8a+6$
- $y^2=(11a+6)x^6+(5a+10)x^5+(12a+1)x^4+6x^3+(2a+4)x^2+(2a+2)x+a+2$
- $y^2=(12a+2)x^6+(11a+6)x^5+(12a+6)x^4+(3a+1)x^3+2ax^2+(a+3)x+2$
- $y^2=(3a+5)x^6+4x^5+(2a+1)x^4+11x^3+(5a+10)x^2+(5a+8)x+a+12$
- $y^2=(6a+7)x^6+(5a+12)x^5+(9a+1)x^4+(a+11)x^3+(9a+10)x^2+(11a+1)x+a+5$
- $y^2=(2a+8)x^6+(10a+12)x^5+(10a+11)x^4+(11a+11)x^3+(12a+11)x^2+(3a+4)x+6a+5$
- $y^2=(a+12)x^6+(2a+3)x^5+(11a+4)x^4+(2a+12)x^3+(3a+7)x^2+(9a+9)x+8a+1$
- $y^2=(6a+9)x^6+(10a+6)x^5+(8a+1)x^4+(5a+2)x^3+(10a+6)x^2+(3a+2)x+10a$
- $y^2=6x^6+8ax^5+7x^4+6ax^3+(4a+3)x^2+(12a+2)x+4a+4$
- $y^2=(9a+12)x^6+(6a+1)x^5+(12a+9)x^4+(10a+5)x^3+(4a+4)x^2+(6a+1)x+4a+1$
- $y^2=(10a+11)x^6+(11a+1)x^5+ax^4+(5a+6)x^3+(4a+6)x^2+(9a+12)x+9a+11$
- $y^2=(6a+12)x^6+(2a+2)x^5+(4a+4)x^4+(3a+10)x^3+2x^2+(10a+11)x+4a+1$
- $y^2=(a+5)x^6+(a+7)x^5+(6a+7)x^4+(6a+8)x^3+(2a+2)x^2+(11a+6)x+8a+8$
- $y^2=2x^6+(10a+1)x^5+(5a+12)x^4+(4a+3)x^3+(4a+8)x^2+(3a+1)x+9a+5$
- $y^2=(12a+12)x^6+(12a+5)x^5+(12a+6)x^4+(7a+4)x^3+(4a+6)x^2+(11a+11)x+12a+7$
- $y^2=(2a+12)x^6+(4a+7)x^5+(6a+4)x^4+(6a+10)x^3+(4a+5)x^2+(10a+4)x+5a+11$
- and 25 more
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 4.0.4901.1. |
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.bt_bgj | $2$ | (not in LMFDB) |