Properties

 Label 2.49.av_hw 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: $1 - 21 x + 204 x^{2} - 1029 x^{3} + 2401 x^{4}$ Frobenius angles: $\pm0.145558870031$, $\pm0.294073133631$ Angle rank: $2$ (numerical) Number field: 4.0.344488.2 Galois group: $D_{4}$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1556 5688736 13900731392 33263563973248 79799379375799316 191581828815879307264 459986553392529725337428 1104427836212166485104200192 2651730978154050392541855822848 6366805811477623273783253635485856

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 29 2369 118154 5770113 282500429 13841330366 678223097693 33232935443329 1628413679151722 79792266931365089

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The endomorphism algebra of this simple isogeny class is 4.0.344488.2.
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.v_hw $2$ (not in LMFDB)