# Properties

 Label 2.49.av_hy 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: $( 1 - 12 x + 49 x^{2} )( 1 - 9 x + 49 x^{2} )$ Frobenius angles: $\pm0.172237328522$, $\pm0.277748883973$ Angle rank: $2$ (numerical) Jacobians: 8

This isogeny class is not 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 8 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1558 5699164 13915663384 33274455030720 79804125241799398 191582843392417555456 459986328472585187815798 1104427523945993268222017280 2651730825137491182999203339224 6366805766422581908047864110941404

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 29 2373 118280 5772001 282517229 13841403666 678222766061 33232926047041 1628413585185080 79792266366710853

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.am $\times$ 1.49.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ad_ak $2$ (not in LMFDB) 2.49.d_ak $2$ (not in LMFDB) 2.49.v_hy $2$ (not in LMFDB)