# Properties

 Label 2.49.ax_iu 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: $( 1 - 13 x + 49 x^{2} )( 1 - 10 x + 49 x^{2} )$ Frobenius angles: $\pm0.121037718324$, $\pm0.246751714429$ Angle rank: $2$ (numerical) Jacobians: 9

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 9 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1480 5594400 13863035680 33259222684800 79802395849731400 191583555362580326400 459986795973662791903240 1104427673621701586376307200 2651730871079049634286196796960 6366805791903204663020946231060000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 27 2329 117834 5769361 282511107 13841455102 678223455363 33232930550881 1628413613397546 79792266686047849

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.an $\times$ 1.49.ak 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.ad_abg $2$ (not in LMFDB) 2.49.d_abg $2$ (not in LMFDB) 2.49.x_iu $2$ (not in LMFDB) 2.49.ai_da $3$ (not in LMFDB) 2.49.b_am $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.ad_abg $2$ (not in LMFDB) 2.49.d_abg $2$ (not in LMFDB) 2.49.x_iu $2$ (not in LMFDB) 2.49.ai_da $3$ (not in LMFDB) 2.49.b_am $3$ (not in LMFDB) 2.49.av_ia $6$ (not in LMFDB) 2.49.am_eo $6$ (not in LMFDB) 2.49.ab_am $6$ (not in LMFDB) 2.49.i_da $6$ (not in LMFDB) 2.49.m_eo $6$ (not in LMFDB) 2.49.v_ia $6$ (not in LMFDB)