# Properties

 Label 2.49.ay_ji 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: $( 1 - 12 x + 49 x^{2} )^{2}$ Frobenius angles: $\pm0.172237328522$, $\pm0.172237328522$ Angle rank: $1$ (numerical) Jacobians: 5

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 5 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1444 5550736 13849994596 33263917830144 79809480702695524 191587709240781609616 459988320399300514995364 1104427961044595342275313664 2651730804630061440313388028196 6366805702924223586474522587602576

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 26 2310 117722 5770174 282536186 13841755206 678225703034 33232939199614 1628413572591578 79792265570914950

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.am 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-13})$$$)$
All geometric endomorphisms are defined over $\F_{7^{2}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{2}}$.

 Subfield Primitive Model $\F_{7}$ 2.7.a_am

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.a_abu $2$ (not in LMFDB) 2.49.y_ji $2$ (not in LMFDB) 2.49.m_dr $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.a_abu $2$ (not in LMFDB) 2.49.y_ji $2$ (not in LMFDB) 2.49.m_dr $3$ (not in LMFDB) 2.49.a_bu $4$ (not in LMFDB) 2.49.am_dr $6$ (not in LMFDB)