Properties

 Label 2.81.abc_nl Base Field $\F_{3^{4}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $( 1 - 17 x + 81 x^{2} )( 1 - 11 x + 81 x^{2} )$ Frobenius angles: $\pm0.106600758076$, $\pm0.290722850198$ Angle rank: $2$ (numerical) Jacobians: 56

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 56 curves, and hence is principally polarizable:

• $y^2=(a^3+2a^2+2a)x^6+2a^2x^5+(a^3+a)x^4+(2a^3+a^2+a+2)x^3+(a^3+a)x^2+2a^2x+a^3+2a^2+2a$
• $y^2=2x^6+(2a^3+1)x^5+(a^3+2a^2+a)x^4+(2a^3+a^2+2a+2)x^3+2a^2x^2+(2a^3+a+1)x+2a^3+2a^2+a+2$
• $y^2=ax^6+(2a^3+a^2+1)x^5+x^4+(2a^2+2)x^3+x^2+(2a^3+a^2+1)x+a$
• $y^2=(2a^3+2a+2)x^6+2ax^5+(a^2+a+2)x^4+(a^3+a^2+2)x^3+(a^2+a+2)x^2+2ax+2a^3+2a+2$
• $y^2=(2a^3+2a^2+a+1)x^6+a^2x^5+(a^3+2a+2)x^4+(a^3+a^2)x^3+(a^3+2a+2)x^2+a^2x+2a^3+2a^2+a+1$
• $y^2=(a^3+a)x^6+(2a^2+2a)x^5+(2a^2+a+2)x^4+(2a^3+2a^2+2a+1)x^3+(a^3+a^2+a+2)x^2+(a^3+2a+2)x+2a^2+2a$
• $y^2=(2a^2+a+2)x^6+(2a^3+a+1)x^5+ax^4+2a^3x^3+ax^2+(2a^3+a+1)x+2a^2+a+2$
• $y^2=(a^3+2a^2+2a+1)x^6+(2a^2+2)x^5+(2a^3+2a^2+2a+1)x^4+(2a^3+a^2+1)x^3+2x^2+(2a^2+a)x+2a^2+2a+2$
• $y^2=(a^3+2a^2+2a+2)x^6+(2a^3+a^2+2a+1)x^5+(a^2+a+1)x^4+(2a^3+a^2+2a)x^3+(2a^3+a^2+1)x^2+2x+a^3+1$
• $y^2=(2a^3+a+2)x^6+2a^3x^5+(a^3+2a^2+a)x^4+(2a^3+2a^2+1)x^3+(a^2+a)x^2+(a^2+2a)x+2a^3$
• $y^2=(2a^3+2a^2+a+1)x^6+(2a^2+a)x^5+(2a^3+a+2)x^4+(a^3+1)x^3+2a^3x^2+(2a^3+2a^2+a+2)x+a^3+2a^2$
• $y^2=(2a^3+a^2)x^6+(a^3+a^2+a+2)x^5+(2a^3+2a)x^4+(2a^2+2a+1)x^3+(2a^3+2a)x^2+(a^3+a^2+a+2)x+2a^3+a^2$
• $y^2=(2a^3+a^2+2a)x^6+(2a^2+2)x^5+(a^3+a^2+2a)x^4+(a^3+a+1)x^3+(2a^3+a^2+1)x^2+(a^2+2a+2)x+a^3+2a^2$
• $y^2=(2a^3+2a+2)x^6+2ax^5+(2a^3+2a^2+2)x^4+(a^2+a+1)x^3+(2a^3+a^2+2)x^2+(2a^3+2)x+2a^3+2a^2$
• $y^2=(2a^3+2a+2)x^6+(a^2+a+1)x^5+(a^3+a^2)x^4+(2a^3+a+2)x^3+(a^3+a^2)x^2+(a^2+a+1)x+2a^3+2a+2$
• $y^2=a^3x^6+(a^3+2a^2+a)x^5+(2a^2+2a+2)x^4+(a+1)x^3+(2a^3+2)x^2+(a^2+2a+1)x+a$
• $y^2=(2a^3+2a^2+a)x^6+x^5+(2a^3+1)x^4+(a^2+2)x^3+(2a^2+a)x^2+(2a^3+a^2+2)x+2a^2$
• $y^2=(a^3+2a^2+a)x^6+(a^3+a)x^4+ax^3+(a^3+a)x^2+a^3+2a^2+a$
• $y^2=(a^3+a^2+1)x^6+(2a^2+a+1)x^5+(2a^3+a+2)x^4+(2a^3+a)x^3+(a^2+2a)x^2+(a^3+1)x+a^2+2a$
• $y^2=(a^3+a+2)x^6+(2a^3+2a^2+2a+2)x^5+(2a^2+1)x^4+(2a^3+a^2+2a+1)x^3+(2a^2+1)x^2+(2a^3+2a^2+2a+2)x+a^3+a+2$
• and 36 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4615 42490305 282727157440 1853383296595545 12157767880071565375 79766362097777811456000 523347568401593656749553135 3433683880049363757802715577705 22528399701917306295297609517831360 147808829578518754283444671557294512625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 54 6476 532002 43055156 3486813774 282429249758 22876789630014 1853020221100196 150094636342860162 12157665472560576476

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.ar $\times$ 1.81.al and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ag_az $2$ (not in LMFDB) 2.81.g_az $2$ (not in LMFDB) 2.81.bc_nl $2$ (not in LMFDB)