Properties

Label 1.361.abm
Base Field $\F_{19^{2}}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{19^{2}}$
Dimension:  $1$
L-polynomial:  $( 1 - 19 x )^{2}$
Frobenius angles:  $0$, $0$
Angle rank:  $0$ (numerical)
Number field:  \(\Q\)
Galois group:  Trivial
Jacobians:  2

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 324 129600 47032164 16983302400 6131061305604 2213314824974400 799006683995140644 288441413533654041600 104127350297265866137284 37589973457533696060840000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 324 129600 47032164 16983302400 6131061305604 2213314824974400 799006683995140644 288441413533654041600 104127350297265866137284 37589973457533696060840000

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19^{2}}$
The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $19$ and $\infty$.
All geometric endomorphisms are defined over $\F_{19^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.361.bm$2$(not in LMFDB)
1.361.a$4$(not in LMFDB)