Properties

 Label 2.49.ax_iv Base Field $\F_{7^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{7^{2}}$ Dimension: $2$ L-polynomial: $1 - 23 x + 229 x^{2} - 1127 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.142622734692$, $\pm0.234081986531$ Angle rank: $2$ (numerical) Number field: 4.0.81125.1 Galois group: $D_{4}$ Jacobians: 4

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

• $y^2=(6a+1)x^6+(a+4)x^5+(3a+2)x^4+4x^3+(5a+2)x^2+ax+3a+3$
• $y^2=(a+1)x^6+(6a+6)x^5+3ax^4+(2a+6)x^3+(2a+5)x^2+6ax+6a+1$
• $y^2=5x^6+(3a+4)x^5+(a+1)x^4+3ax^3+(3a+5)x^2+6x+5a+1$
• $y^2=(5a+6)x^6+(5a+6)x^5+(5a+5)x^4+(a+6)x^3+6ax^2+6x+4a+3$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1481 5599661 13871225201 33266158087445 79806326823818496 191585122056539279501 459987182520552743279801 1104427667528199114470953445 2651730804376104684256767464081 6366805749648344848092123125846016

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 27 2331 117903 5770563 282525022 13841568291 678224025303 33232930367523 1628413572435627 79792266156487006

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.81125.1.
All geometric endomorphisms are defined over $\F_{7^{2}}$.

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 2.49.x_iv $2$ (not in LMFDB)