# Properties

 Label 2.49.av_hu 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: $( 1 - 13 x + 49 x^{2} )( 1 - 8 x + 49 x^{2} )$ Frobenius angles: $\pm0.121037718324$, $\pm0.306389419005$ Angle rank: $2$ (numerical) Jacobians: 16

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 16 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1554 5678316 13885804296 33252581908224 79794396406737474 191580510300991209216 459986523882187819594434 1104427996894746505434184704 2651731067238129001644603234696 6366805840668972677984960081957676

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 29 2365 118028 5768209 282482789 13841235106 678223054181 33232940278369 1628413733857772 79792267297206925

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.an $\times$ 1.49.ai 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 are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.af_ag $2$ (not in LMFDB) 2.49.f_ag $2$ (not in LMFDB) 2.49.v_hu $2$ (not in LMFDB) 2.49.ag_de $3$ (not in LMFDB) 2.49.d_k $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.af_ag $2$ (not in LMFDB) 2.49.f_ag $2$ (not in LMFDB) 2.49.v_hu $2$ (not in LMFDB) 2.49.ag_de $3$ (not in LMFDB) 2.49.d_k $3$ (not in LMFDB) 2.49.at_he $6$ (not in LMFDB) 2.49.ak_ek $6$ (not in LMFDB) 2.49.ad_k $6$ (not in LMFDB) 2.49.g_de $6$ (not in LMFDB) 2.49.k_ek $6$ (not in LMFDB) 2.49.t_he $6$ (not in LMFDB)