# Properties

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

## Invariants

 Base field: $\F_{7^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 7 x )^{2}( 1 - 8 x + 49 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.306389419005$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1512 5612544 13838478696 33226260480000 79781820476078952 191575128888463574016 459984415854857021776872 1104427232468159779307520000 2651730809531246043410493784296 6366805760003910324030652455650304

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 28 2338 117628 5763646 282438268 13840846306 678219946012 33232917276286 1628413575601372 79792266286268578

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.ao $\times$ 1.49.ai 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.ai : $$\Q(\sqrt{-33})$$.
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.ag_ao $2$ (not in LMFDB) 2.49.g_ao $2$ (not in LMFDB) 2.49.w_ic $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.ag_ao $2$ (not in LMFDB) 2.49.g_ao $2$ (not in LMFDB) 2.49.w_ic $2$ (not in LMFDB) 2.49.ai_du $4$ (not in LMFDB) 2.49.i_du $4$ (not in LMFDB)