Properties

Label 3.6.3.24a
Base 3.1.1.0a1.1
Degree \(18\)
e \(3\)
f \(6\)
c \(24\)

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Defining polynomial over unramified subextension

$x^{3} + 3 a_{2} x^{2} + 9 c_{3} + 3$

Invariants

Residue field characteristic: $3$
Degree: $18$
Base field: $\Q_{3}$
Ramification index $e$: $3$
Residue field degree $f$: $6$
Discriminant exponent $c$: $24$
Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{2}{3}\rangle$
Rams: $(1)$
Field count: $258$ (complete)
Ambiguity: $18$
Mass: $728$
Absolute Mass: $364/3$

Diagrams

Varying

Indices of inseparability: $[2,0]$
Associated inertia: $[1]$ (show 196), $[2]$ (show 62)
Jump Set: undefined

Fields


Showing 1-50 of 258

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Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
3.6.3.24a1.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 3 ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $S_3 \times C_3$ (as 18T3) $18$ $18$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^3 + 2)$ undefined
3.6.3.24a1.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 3 ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 9 x + 3$ $C_9\times S_3$ (as 18T16) $54$ $9$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^3 + 2)$ undefined
3.6.3.24a2.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 6 ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $18$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^3 + 1)$ undefined
3.6.3.24a2.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 6 ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 9 x^{2} + 3$ $C_{18}$ (as 18T1) $18$ $18$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^3 + 1)$ undefined
3.6.3.24a2.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 6 ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 18 x^{2} + 3$ $C_{18}$ (as 18T1) $18$ $18$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^3 + 1)$ undefined
3.6.3.24a3.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 3 x ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[2, 0]$ $[2]$ $z^2 + (2 t^4 + 2 t)$ undefined
3.6.3.24a4.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 6 x ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[2, 0]$ $[2]$ $z^2 + (t^4 + t)$ undefined
3.6.3.24a5.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 3 x^{2} ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + 2 t^2)$ undefined
3.6.3.24a5.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 3 x^{2} ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 9 x + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + 2 t^2)$ undefined
3.6.3.24a5.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 3 x^{2} ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 18 x + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + 2 t^2)$ undefined
3.6.3.24a6.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{2} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[2, 0]$ $[2]$ $z^2 + (2 t^5 + 2 t^4 + t^3 + 2 t^2 + 2 t + 1)$ undefined
3.6.3.24a7.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 3 x + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $C_3^4:C_6$ (as 18T163) $486$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^4 + t^3 + t^2 + t + 1)$ undefined
3.6.3.24a7.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 3 x + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^4 + t^3 + t^2 + t + 1)$ undefined
3.6.3.24a7.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 3 x + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 21$ $C_3^4:C_6$ (as 18T163) $486$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^4 + t^3 + t^2 + t + 1)$ undefined
3.6.3.24a8.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 3 x^{2} + 6 x\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + 2 t^3 + 2)$ undefined
3.6.3.24a8.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 3 x^{2} + 6 x\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + 2 t^3 + 2)$ undefined
3.6.3.24a8.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 3 x^{2} + 6 x\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 18 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + 2 t^3 + 2)$ undefined
3.6.3.24a9.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 6 x^{2} + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^5 + 2 t^4 + t^3 + 2 t^2 + 2 t + 1)$ undefined
3.6.3.24a9.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 6 x^{2} + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 12$ $C_3^5:C_6$ (as 18T323) $1458$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^5 + 2 t^4 + t^3 + 2 t^2 + 2 t + 1)$ undefined
3.6.3.24a9.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 6 x^{2} + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 21$ $C_3^5:C_6$ (as 18T323) $1458$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^5 + 2 t^4 + t^3 + 2 t^2 + 2 t + 1)$ undefined
3.6.3.24a10.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{3} + 6 x^{2} + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $C_3^4:C_{12}$ (as 18T240) $972$ $1$ not computed not computed not computed not computed $[2, 0]$ $[2]$ $z^2 + (t^5 + 2 t^4 + 2 t^2 + 2 t)$ undefined
3.6.3.24a11.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(6 x^{3} + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $C_3^5:C_6$ (as 18T323) $1458$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^4 + 2 t^3 + 2 t^2 + t + 2)$ undefined
3.6.3.24a11.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(6 x^{3} + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^4 + 2 t^3 + 2 t^2 + t + 2)$ undefined
3.6.3.24a11.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(6 x^{3} + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 21$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^4 + 2 t^3 + 2 t^2 + t + 2)$ undefined
3.6.3.24a12.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(6 x^{3} + 3 x\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^3 + 2 t^2 + 1)$ undefined
3.6.3.24a12.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(6 x^{3} + 3 x\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^3 + 2 t^2 + 1)$ undefined
3.6.3.24a12.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(6 x^{3} + 3 x\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 21$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (t^3 + 2 t^2 + 1)$ undefined
3.6.3.24a13.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(6 x^{3} + 3 x^{2} + 6 x\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $C_3^4:C_{12}$ (as 18T240) $972$ $1$ not computed not computed not computed not computed $[2, 0]$ $[2]$ $z^2 + (2 t^5 + 2 t^4 + t^3 + t^2 + 2 t + 1)$ undefined
3.6.3.24a14.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(6 x^{3} + 6 x^{2} + 6 x\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[2, 0]$ $[2]$ $z^2 + (t^5 + 2 t^4 + t^3 + 2 t + 1)$ undefined
3.6.3.24a15.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(6 x^{3} + 6 x^{2} + 6 x + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[2, 0]$ $[2]$ $z^2 + (t^5 + 2 t^4 + 2 t)$ undefined
3.6.3.24a16.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 3 x^{4} ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + 2 t^4 + t^3 + 2 t^2 + 2 t)$ undefined
3.6.3.24a16.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 3 x^{4} ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 12$ $C_3^5:C_6$ (as 18T323) $1458$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + 2 t^4 + t^3 + 2 t^2 + 2 t)$ undefined
3.6.3.24a16.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + 3 x^{4} ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 21$ $C_3^5:C_6$ (as 18T323) $1458$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + 2 t^4 + t^3 + 2 t^2 + 2 t)$ undefined
3.6.3.24a17.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $C_3\wr C_6$ (as 18T453) $4374$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^4 + 2 t^3 + 2 t^2 + t + 1)$ undefined
3.6.3.24a17.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^4 + 2 t^3 + 2 t^2 + t + 1)$ undefined
3.6.3.24a17.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 21$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^4 + 2 t^3 + 2 t^2 + t + 1)$ undefined
3.6.3.24a18.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 3 x^{3} + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $C_3\wr C_6$ (as 18T453) $4374$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^4 + 2 t^3 + t + 1)$ undefined
3.6.3.24a18.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 3 x^{3} + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^4 + 2 t^3 + t + 1)$ undefined
3.6.3.24a18.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 3 x^{3} + 3\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 21$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^4 + 2 t^3 + t + 1)$ undefined
3.6.3.24a19.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 3 x^{3} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $C_3^4:C_6$ (as 18T163) $486$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^3)$ undefined
3.6.3.24a19.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 3 x^{3} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^3)$ undefined
3.6.3.24a19.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 3 x^{3} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 21$ $C_3^4:C_6$ (as 18T163) $486$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^3)$ undefined
3.6.3.24a20.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 6 x^{3} + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $C_3^3:C_{18}$ (as 18T159) $486$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^2 + 2)$ undefined
3.6.3.24a20.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 6 x^{3} + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 9 x^{2} + 3$ $C_3^3:C_6$ (as 18T85) $162$ $6$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^2 + 2)$ undefined
3.6.3.24a21.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 6 x^{3} + 6 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^4 + t^2 + t + 2)$ undefined
3.6.3.24a21.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 6 x^{3} + 6 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^4 + t^2 + t + 2)$ undefined
3.6.3.24a21.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 6 x^{3} + 6 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 21$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^5 + t^4 + t^2 + t + 2)$ undefined
3.6.3.24a22.1 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 3$ $C_3^5:C_6$ (as 18T327) $1458$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^4 + 2 t^2 + 2 t + 2)$ undefined
3.6.3.24a22.2 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 9 x + 3$ $C_3^5.C_6$ (as 18T325) $1458$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^4 + 2 t^2 + 2 t + 2)$ undefined
3.6.3.24a22.3 $( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{3} + \left(3 x^{4} + 6 x^{3} + 6 x^{2} + 3 x + 6\right) ( x^{6} + 2 x^{4} + x^{2} + 2 x + 2 )^{2} + 18 x + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z^2 + (2 t^4 + 2 t^2 + 2 t + 2)$ undefined
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