Properties

 Label 2.49.ax_ir 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: $1 - 23 x + 225 x^{2} - 1127 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.0550320885809$, $\pm0.271501929914$ Angle rank: $2$ (numerical) Number field: 4.0.284445.2 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1477 5578629 13838475301 33238280483205 79790213515005952 191578291100868522501 459985079447809912612357 1104427274187618598029610245 2651730820716027483375496370341 6366805796193014813228463108194304

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 27 2323 117627 5765731 282467982 13841074771 678220924443 33232918531651 1628413582469883 79792266739810078

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.284445.2.
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_ir $2$ (not in LMFDB)