# Properties

 Label 2.49.aw_ih 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: $( 1 - 13 x + 49 x^{2} )( 1 - 9 x + 49 x^{2} )$ Frobenius angles: $\pm0.121037718324$, $\pm0.277748883973$ 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1517 5638689 13877588816 33257579784345 79798599518267477 191581729178259603456 459986427486813036231077 1104427761809799742219228905 2651730972034437890615161956176 6366805831001365816387225445552529

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 28 2348 117958 5769076 282497668 13841323166 678222912052 33232933204516 1628413675393702 79792267176047228

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.an $\times$ 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 are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.ae_at $2$ (not in LMFDB) 2.49.e_at $2$ (not in LMFDB) 2.49.w_ih $2$ (not in LMFDB) 2.49.ah_dc $3$ (not in LMFDB) 2.49.c_ab $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.ae_at $2$ (not in LMFDB) 2.49.e_at $2$ (not in LMFDB) 2.49.w_ih $2$ (not in LMFDB) 2.49.ah_dc $3$ (not in LMFDB) 2.49.c_ab $3$ (not in LMFDB) 2.49.au_hp $6$ (not in LMFDB) 2.49.al_em $6$ (not in LMFDB) 2.49.ac_ab $6$ (not in LMFDB) 2.49.h_dc $6$ (not in LMFDB) 2.49.l_em $6$ (not in LMFDB) 2.49.u_hp $6$ (not in LMFDB)