# Properties

 Label 2.7.ae_i 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: $1 - 4 x + 8 x^{2} - 28 x^{3} + 49 x^{4}$ Frobenius angles: $\pm0.0704914820143$, $\pm0.570491482014$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{10})$$ Galois group: $C_2^2$ Jacobians: 3

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 3 curves, and hence is principally polarizable:

• $y^2=x^6+6x^5+x^4+2x^3+6x^2+x$
• $y^2=x^6+4x^5+5x^4+5x^2+3x+1$
• $y^2=2x^5+2x^4+2x^3+6x^2+4x+3$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 26 2340 101114 5475600 284431706 13841495460 676699584314 33243988377600 1629091518163226 79792266455518500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 50 292 2278 16924 117650 821692 5766718 40370404 282475250

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{10})$$.
Endomorphism algebra over $\overline{\F}_{7}$
 The base change of $A$ to $\F_{7^{4}}$ is 1.2401.ack 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-10})$$$)$
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_ack and its endomorphism algebra is $$\Q(i, \sqrt{10})$$.

## 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 2.7.e_i $2$ 2.49.a_ack 2.7.e_i $4$ (not in LMFDB) 2.7.a_ag $8$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.7.e_i $2$ 2.49.a_ack 2.7.e_i $4$ (not in LMFDB) 2.7.a_ag $8$ (not in LMFDB) 2.7.a_g $8$ (not in LMFDB)