# Properties

 Label 2.49.au_ha Base Field $\F_{7^{2}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{7^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 7 x )^{2}( 1 - 6 x + 49 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.359017035971$ Angle rank: $1$ (numerical) Jacobians: 6

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1584 5677056 13838712624 33210777600000 79775178013683504 191575092071265509376 459985465482543519807024 1104427643743877470617600000 2651730812702512974796073358384 6366805703230630314611468677718016

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 30 2366 117630 5760958 282414750 13840843646 678221493630 33232929651838 1628413577548830 79792265574755006

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.ao $\times$ 1.49.ag and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.49.ao : the quaternion algebra over $$\Q$$ ramified at $7$ and $\infty$. 1.49.ag : $$\Q(\sqrt{-10})$$.
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.ai_o $2$ (not in LMFDB) 2.49.i_o $2$ (not in LMFDB) 2.49.u_ha $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.ai_o $2$ (not in LMFDB) 2.49.i_o $2$ (not in LMFDB) 2.49.u_ha $2$ (not in LMFDB) 2.49.ag_du $4$ (not in LMFDB) 2.49.g_du $4$ (not in LMFDB)