# Properties

 Label 2.49.ay_je 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: $( 1 - 7 x )^{2}( 1 - 10 x + 49 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.246751714429$ 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1440 5529600 13815787680 33232896000000 79789818658327200 191578173864518246400 459984687945085052629920 1104426909195338612736000000 2651730613372185739653797292960 6366805711238142926910784650240000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 26 2302 117434 5764798 282466586 13841066302 678220347194 33232907548798 1628413455141146 79792265675109502

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.ao $\times$ 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: 1.49.ao : the quaternion algebra over $$\Q$$ ramified at $7$ and $\infty$. 1.49.ak : $$\Q(\sqrt{-6})$$.
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.ae_abq $2$ (not in LMFDB) 2.49.e_abq $2$ (not in LMFDB) 2.49.y_je $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.ae_abq $2$ (not in LMFDB) 2.49.e_abq $2$ (not in LMFDB) 2.49.y_je $2$ (not in LMFDB) 2.49.ak_du $4$ (not in LMFDB) 2.49.k_du $4$ (not in LMFDB)