Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 45 x + 836 x^{2} - 7605 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.0700871878856$, $\pm0.227679424698$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2552616.2 |
Galois group: | $D_{4}$ |
Jacobians: | $18$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21748$ | $805719904$ | $23292873529600$ | $665431721861791104$ | $19005010271152912121428$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $125$ | $28209$ | $4825730$ | $815749249$ | $137858829125$ | $23298086679918$ | $3937376345565725$ | $665416608110979649$ | $112455406939471434290$ | $19004963774865592960929$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+12)x^6+(2a+2)x^5+(8a+12)x^4+(a+3)x^3+(12a+2)x^2+(7a+2)x+9a+6$
- $y^2=(12a+5)x^6+5ax^5+(4a+10)x^4+(6a+7)x^3+(a+9)x^2+(3a+12)x+9a+6$
- $y^2=(7a+3)x^6+(9a+9)x^5+7x^4+12x^3+8x^2+(4a+8)x+11a+12$
- $y^2=(7a+10)x^6+(10a+3)x^5+(8a+12)x^4+(a+4)x^3+10ax^2+(8a+7)x+3a+10$
- $y^2=(10a+11)x^6+(10a+9)x^5+(3a+10)x^4+(2a+1)x^3+(11a+2)x^2+(2a+11)x+9a+6$
- $y^2=(10a+11)x^6+(9a+7)x^5+(6a+1)x^4+(10a+11)x^3+(2a+11)x^2+(9a+1)x+12a+10$
- $y^2=(6a+10)x^6+(9a+2)x^5+(9a+4)x^4+(a+5)x^3+(5a+9)x^2+(9a+4)x+11a+2$
- $y^2=(2a+4)x^6+(2a+4)x^5+(10a+7)x^4+(11a+11)x^3+(12a+8)x^2+(9a+4)x+6a+12$
- $y^2=7ax^6+(a+9)x^5+10x^4+(7a+1)x^3+5ax^2+(7a+10)x+5a+10$
- $y^2=(8a+11)x^6+(2a+12)x^5+(4a+2)x^4+(11a+2)x^3+(7a+12)x^2+(2a+6)x+5a+11$
- $y^2=9ax^6+(7a+7)x^5+3x^4+9x^3+(3a+2)x^2+(a+11)x+9a+12$
- $y^2=(7a+6)x^6+(11a+9)x^5+(7a+11)x^4+(5a+9)x^3+(12a+7)x^2+(a+9)x+10a+8$
- $y^2=11x^6+(5a+2)x^5+(8a+12)x^4+(7a+7)x^3+(8a+7)x^2+8x+a+11$
- $y^2=(6a+6)x^6+6ax^5+(9a+5)x^4+(10a+6)x^3+(3a+5)x^2+(2a+10)x+7a+1$
- $y^2=2ax^6+3x^5+(8a+1)x^4+(8a+6)x^3+(8a+12)x^2+(7a+2)x+12a+11$
- $y^2=(12a+1)x^6+(3a+12)x^5+(6a+6)x^4+(a+6)x^3+(5a+7)x^2+(10a+2)x+8a$
- $y^2=10ax^6+(10a+11)x^5+(11a+7)x^4+(9a+11)x^3+(a+6)x^2+(8a+9)x+2$
- $y^2=(10a+7)x^6+(11a+11)x^5+(3a+8)x^4+(a+3)x^3+(4a+9)x^2+(6a+5)x+12a+8$
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 4.0.2552616.2. |
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_bge | $2$ | (not in LMFDB) |