Properties

 Label 2.81.abc_nn Base Field $\F_{3^{4}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $1 - 28 x + 351 x^{2} - 2268 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.124262281589$, $\pm0.282730282950$ Angle rank: $2$ (numerical) Number field: 4.0.88592.1 Galois group: $D_{4}$ Jacobians: 4

This isogeny class is simple and geometrically simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4617 42517953 282816754308 1853532787754313 12157928970691398057 79766478695865098248848 523347615455109181447945833 3433683871265097726749572691337 22528399669522869732220049926149828 147808829548276655830696851322697565633

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 54 6480 532170 43058628 3486859974 282429662598 22876791686838 1853020216359684 150094636127033418 12157665470073084240

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.88592.1.
All geometric endomorphisms are defined over $\F_{3^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.bc_nn $2$ (not in LMFDB)