# Properties

 Label 2.7.ag_s Base Field $\F_{7}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{7}$ Dimension: $2$ L-polynomial: $1 - 6 x + 18 x^{2} - 42 x^{3} + 49 x^{4}$ Frobenius angles: $\pm0.0461154155528$, $\pm0.453884584447$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{5})$$ Galois group: $C_2^2$ Jacobians: 2

This isogeny class is simple but not geometrically 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 2 curves, and hence is principally polarizable:

• $y^2=6x^6+4x^5+6x^4+6x^2+3x+6$
• $y^2=4x^6+3x^4+4x^3+5x^2+4x+6$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 20 2320 111620 5382400 276390500 13841326480 678702544820 33210785894400 1627783961277620 79792266845818000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 50 326 2238 16442 117650 824126 5760958 40338002 282475250

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{7}$
 The base change of $A$ to $\F_{7^{4}}$ is 1.2401.ade 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$
All geometric endomorphisms are defined over $\F_{7^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{7^{2}}$  The base change of $A$ to $\F_{7^{2}}$ is the simple isogeny class 2.49.a_ade and its endomorphism algebra is $$\Q(i, \sqrt{5})$$.

## 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 2.7.g_s $2$ 2.49.a_ade 2.7.a_ae $8$ (not in LMFDB) 2.7.a_e $8$ (not in LMFDB)