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Properties

Label 5.3.5.18a
Base 5.1.1.0a1.1
Degree \(15\)
e \(5\)
f \(3\)
c \(18\)

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Defining polynomial over unramified subextension

$x^{5} + 5a_{2} x^{2} + 5$

Invariants

Residue field characteristic: $5$
Degree: $15$
Base field: $\Q_{5}$
Ramification index $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $18$
Artin slopes: $[\frac{3}{2}]$
Swan slopes: $[\frac{1}{2}]$
Means: $\langle\frac{2}{5}\rangle$
Rams: $(\frac{1}{2})$
Field count: $44$ (complete)
Ambiguity: $3$
Mass: $124$
Absolute Mass: $124/3$

Diagrams

Varying

Indices of inseparability: $[2,0]$
Associated inertia: $[1]$ (show 22), $[2]$ (show 22)
Jump Set: undefined

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 44

  displayed columns for results
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
5.3.5.18a1.1 1 $( x^{3} + 3 x + 3 )^{5} + 10 ( x^{3} + 3 x + 3 )^{2} + 5$ $D_5\times C_3$ (as 15T3) $30$ $3$ $[\frac{3}{2}]_{2}^{3}$ $[\frac{1}{2}]_{2}^{3}$ $[\ ]_{2}$ $[\ ]_{2}$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 3 t + 4)$ undefined
5.3.5.18a2.1 1 $( x^{3} + 3 x + 3 )^{5} + 20 ( x^{3} + 3 x + 3 )^{2} + 5$ $F_5\times C_3$ (as 15T8) $60$ $3$ $[\frac{3}{2}]_{2}^{6}$ $[\frac{1}{2}]_{2}^{6}$ $[\ ]^{2}_{2}$ $[\ ]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (2 t^2 + t + 3)$ undefined
5.3.5.18a3.1 1 $( x^{3} + 3 x + 3 )^{5} + 5 ( x^{3} + 3 x + 3 )^{2} + 5$ $F_5\times C_3$ (as 15T8) $60$ $3$ $[\frac{3}{2}]_{2}^{6}$ $[\frac{1}{2}]_{2}^{6}$ $[\ ]^{2}_{2}$ $[\ ]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (3 t^2 + 4 t + 2)$ undefined
5.3.5.18a4.1 1 $( x^{3} + 3 x + 3 )^{5} + 15 ( x^{3} + 3 x + 3 )^{2} + 5$ $D_5\times C_3$ (as 15T3) $30$ $3$ $[\frac{3}{2}]_{2}^{3}$ $[\frac{1}{2}]_{2}^{3}$ $[\ ]_{2}$ $[\ ]_{2}$ $[2, 0]$ $[1]$ $z^2 + (4 t^2 + 2 t + 1)$ undefined
5.3.5.18a5.1 1 $( x^{3} + 3 x + 3 )^{5} + 10 x ( x^{3} + 3 x + 3 )^{2} + 5$ $(C_5^2 : C_4):C_3$ (as 15T19) $300$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (3 t^2 + t + 2)$ undefined
5.3.5.18a6.1 1 $( x^{3} + 3 x + 3 )^{5} + 20 x ( x^{3} + 3 x + 3 )^{2} + 5$ $(C_5^2 : C_3):C_2$ (as 15T12) $150$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 2 t + 4)$ undefined
5.3.5.18a7.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x + 20\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (3 t^2 + 3 t + 2)$ undefined
5.3.5.18a8.1 1 $( x^{3} + 3 x + 3 )^{5} + 5 x ( x^{3} + 3 x + 3 )^{2} + 5$ $(C_5^2 : C_3):C_2$ (as 15T12) $150$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (4 t^2 + 3 t + 1)$ undefined
5.3.5.18a9.1 1 $( x^{3} + 3 x + 3 )^{5} + 15 x ( x^{3} + 3 x + 3 )^{2} + 5$ $(C_5^2 : C_4):C_3$ (as 15T19) $300$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (2 t^2 + 4 t + 3)$ undefined
5.3.5.18a10.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(15 x + 5\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + 3 t$ undefined
5.3.5.18a11.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(15 x + 15\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (t^2 + t + 4)$ undefined
5.3.5.18a12.1 1 $( x^{3} + 3 x + 3 )^{5} + 10 x^{2} ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 3 t + 1)$ undefined
5.3.5.18a13.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(10 x^{2} + 10 x + 10\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (2 t + 2)$ undefined
5.3.5.18a14.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(10 x^{2} + 20 x + 10\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (3 t^2 + 3 t + 4)$ undefined
5.3.5.18a15.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(10 x^{2} + 5 x + 20\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + 2 t)$ undefined
5.3.5.18a16.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(10 x^{2} + 5 x + 5\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (3 t^2 + 4)$ undefined
5.3.5.18a17.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(10 x^{2} + 15 x + 10\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (4 t^2 + 3)$ undefined
5.3.5.18a18.1 1 $( x^{3} + 3 x + 3 )^{5} + 20 x^{2} ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (2 t^2 + t + 2)$ undefined
5.3.5.18a19.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 10 x + 10\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 3)$ undefined
5.3.5.18a20.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 20 x\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (3 t^2 + 3 t + 1)$ undefined
5.3.5.18a21.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 20 x + 5\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (t^2 + 2 t + 3)$ undefined
5.3.5.18a22.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 20 x + 15\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + 2)$ undefined
5.3.5.18a23.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 5 x\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $(C_5^2 : C_4):C_3$ (as 15T19) $300$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (t^2 + 4 t + 3)$ undefined
5.3.5.18a24.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 5 x + 20\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (3 t^2 + 1)$ undefined
5.3.5.18a25.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 5 x + 15\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (t + 4)$ undefined
5.3.5.18a26.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 15 x + 10\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (3 t + 4)$ undefined
5.3.5.18a27.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 15 x + 20\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (t^2 + t + 3)$ undefined
5.3.5.18a28.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 15 x + 5\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (2 t^2 + 4 t + 2)$ undefined
5.3.5.18a29.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 20\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + 1$ undefined
5.3.5.18a30.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 10 x + 5\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (4 t^2 + 4 t + 2)$ undefined
5.3.5.18a31.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 20 x\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $(C_5^2 : C_4):C_3$ (as 15T19) $300$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (4 t^2 + t + 2)$ undefined
5.3.5.18a32.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 20 x + 5\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (2 t^2 + 4)$ undefined
5.3.5.18a33.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 5 x + 5\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (t + 1)$ undefined
5.3.5.18a34.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 5 x + 15\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $(C_5^2 : C_3):C_2$ (as 15T12) $150$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 4 t)$ undefined
5.3.5.18a35.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 15 x + 10\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (t^2 + t)$ undefined
5.3.5.18a36.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 15 x + 20\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (2 t^2 + 4 t + 4)$ undefined
5.3.5.18a37.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 15 x + 15\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (4 t^2 + 2)$ undefined
5.3.5.18a38.1 1 $( x^{3} + 3 x + 3 )^{5} + 15 x^{2} ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (4 t^2 + 2 t + 4)$ undefined
5.3.5.18a39.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(15 x^{2} + 10\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + 3$ undefined
5.3.5.18a40.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(15 x^{2} + 15\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (3 t^2 + 4 t)$ undefined
5.3.5.18a41.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(15 x^{2} + 10 x\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $(C_5^2 : C_3):C_2$ (as 15T12) $150$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + 3 t + 1)$ undefined
5.3.5.18a42.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(15 x^{2} + 20 x + 10\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 2 t + 2)$ undefined
5.3.5.18a43.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(15 x^{2} + 5 x + 20\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_{12}$ (as 15T38) $1500$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{6}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{6}$ $[\frac{3}{2},\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{1}{2}]^{2}_{2}$ $[2, 0]$ $[2]$ $z^2 + (t + 3)$ undefined
5.3.5.18a44.1 1 $( x^{3} + 3 x + 3 )^{5} + \left(15 x^{2} + 5 x + 15\right) ( x^{3} + 3 x + 3 )^{2} + 5$ $C_5^3:C_6$ (as 15T30) $750$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + 2 t + 1)$ undefined
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