pp-adic field search results

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Label	$n$	Polynomial	$p$	$f$	$e$	$c$	Galois group	$u$	$t$	Visible Artin slopes	Visible Swan slopes	Artin slope content	Swan slope content	Hidden Artin slopes	Hidden Swan slopes	$\#\Aut(K/\Q_p)$	Unram. Ext.	Eisen. Poly.	Ind. of Insep.	Assoc. Inertia	Resid. Poly	Jump Set
7.14.1.0a1.1	$14$	$x^{14} + 5 x^{7} + 6 x^{5} + 2 x^{4} + 3 x^{2} + 6 x + 3$	$7$	$14$	$1$	$0$	$C_{14}$ (as 14T1)	$14$	$1$	$[\ ]$	$[\ ]$	$[\ ]^{14}$	$[\ ]^{14}$	$[\ ]$	$[\ ]$	$14$	$t^{14} + 5 t^{7} + 6 t^{5} + 2 t^{4} + 3 t^{2} + 6 t + 3$	$x - 7$	$[0]$	$[\ ]$		undefined
7.7.2.7a1.1	$14$	$( x^{7} + 6 x + 4 )^{2} + 7 x$	$7$	$7$	$2$	$7$	$C_{14}$ (as 14T1)	$7$	$2$	$[\ ]$	$[\ ]$	$[\ ]_{2}^{7}$	$[\ ]_{2}^{7}$	$[\ ]$	$[\ ]$	$14$	$t^{7} + 6 t + 4$	$x^{2} + 7 t$	$[0]$	$[1]$	$z + 2$	undefined
7.7.2.7a1.2	$14$	$( x^{7} + 6 x + 4 )^{2} + 7$	$7$	$7$	$2$	$7$	$C_{14}$ (as 14T1)	$7$	$2$	$[\ ]$	$[\ ]$	$[\ ]_{2}^{7}$	$[\ ]_{2}^{7}$	$[\ ]$	$[\ ]$	$14$	$t^{7} + 6 t + 4$	$x^{2} + 7$	$[0]$	$[1]$	$z + 2$	undefined
7.2.7.14a1.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 42 ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}]_{6}^{2}$	$[\frac{1}{6}]_{6}^{2}$	$[\ ]_{6}$	$[\ ]_{6}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 42 x + 7$	$[1, 0]$	$[1]$	$z + (5 t + 1)$	undefined
7.2.7.14a2.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 35 ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}]_{6}^{2}$	$[\frac{1}{6}]_{6}^{2}$	$[\ ]_{6}$	$[\ ]_{6}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 35 x + 7$	$[1, 0]$	$[1]$	$z + (3 t + 2)$	undefined
7.2.7.14a3.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 28 ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}]_{6}^{2}$	$[\frac{1}{6}]_{6}^{2}$	$[\ ]_{6}$	$[\ ]_{6}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 28 x + 7$	$[1, 0]$	$[1]$	$z + (t + 3)$	undefined
7.2.7.14a4.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 21 ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}]_{6}^{2}$	$[\frac{1}{6}]_{6}^{2}$	$[\ ]_{6}$	$[\ ]_{6}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 21 x + 7$	$[1, 0]$	$[1]$	$z + (6 t + 4)$	undefined
7.2.7.14a5.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 14 ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}]_{6}^{2}$	$[\frac{1}{6}]_{6}^{2}$	$[\ ]_{6}$	$[\ ]_{6}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 14 x + 7$	$[1, 0]$	$[1]$	$z + (4 t + 5)$	undefined
7.2.7.14a6.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 7 ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$F_7 \times C_2$ (as 14T7)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}]_{6}^{2}$	$[\frac{1}{6}]_{6}^{2}$	$[\ ]_{6}$	$[\ ]_{6}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 7 x + 7$	$[1, 0]$	$[1]$	$z + (2 t + 6)$	undefined
7.2.7.14a7.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 42 x ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 42\right) x + 7$	$[1, 0]$	$[1]$	$z + (6 t + 2)$	undefined
7.2.7.14a8.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(42 x + 42\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 35\right) x + 7$	$[1, 0]$	$[1]$	$z + (4 t + 3)$	undefined
7.2.7.14a9.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(42 x + 21\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 14\right) x + 7$	$[1, 0]$	$[1]$	$z + (5 t + 6)$	undefined
7.2.7.14a10.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(42 x + 14\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 7\right) x + 7$	$[1, 0]$	$[1]$	$z + 3 t$	undefined
7.2.7.14a11.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 35 x ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + 35 t x + 7$	$[1, 0]$	$[1]$	$z + (5 t + 4)$	undefined
7.2.7.14a12.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(35 x + 42\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 28\right) x + 7$	$[1, 0]$	$[1]$	$z + (3 t + 5)$	undefined
7.2.7.14a13.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(35 x + 28\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(35 t + 28\right) x + 7$	$[1, 0]$	$[1]$	$z + 6 t$	undefined
7.2.7.14a14.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(35 x + 21\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(35 t + 21\right) x + 7$	$[1, 0]$	$[1]$	$z + (4 t + 1)$	undefined
7.2.7.14a15.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(35 x + 7\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 42\right) x + 7$	$[1, 0]$	$[1]$	$z + 3$	undefined
7.2.7.14a16.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 28 x ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(21 t + 28\right) x + 7$	$[1, 0]$	$[1]$	$z + (4 t + 6)$	undefined
7.2.7.14a17.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(28 x + 42\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(28 t + 42\right) x + 7$	$[1, 0]$	$[1]$	$z + 2 t$	undefined
7.2.7.14a18.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(28 x + 28\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(21 t + 7\right) x + 7$	$[1, 0]$	$[1]$	$z + (5 t + 2)$	undefined
7.2.7.14a19.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(28 x + 14\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(28 t + 14\right) x + 7$	$[1, 0]$	$[1]$	$z + (t + 4)$	undefined
7.2.7.14a20.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 21 x ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + 21 t x + 7$	$[1, 0]$	$[1]$	$z + (3 t + 1)$	undefined
7.2.7.14a21.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(21 x + 35\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(21 t + 35\right) x + 7$	$[1, 0]$	$[1]$	$z + (6 t + 3)$	undefined
7.2.7.14a22.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(21 x + 14\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(28 t + 35\right) x + 7$	$[1, 0]$	$[1]$	$z + 6$	undefined
7.2.7.14a23.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 14 x ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + 14 t x + 7$	$[1, 0]$	$[1]$	$z + (2 t + 3)$	undefined
7.2.7.14a24.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(14 x + 21\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(35 t + 35\right) x + 7$	$[1, 0]$	$[1]$	$z + t$	undefined
7.2.7.14a25.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 7 x ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$C_7^2:C_{12}$ (as 14T23)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + 7 t x + 7$	$[1, 0]$	$[1]$	$z + (t + 5)$	undefined
7.2.7.14a26.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(7 x + 28\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(42 t + 35\right) x + 7$	$[1, 0]$	$[1]$	$z + (2 t + 1)$	undefined
7.2.7.14a27.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(7 x + 21\right) ( x^{2} + 6 x + 3 ) + 7$	$7$	$2$	$7$	$14$	$D_7:F_7$ (as 14T24)	$2$	$6$	$[\frac{7}{6}]$	$[\frac{1}{6}]$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]_{6}$	$[\frac{1}{6}]_{6}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(42 t + 28\right) x + 7$	$[1, 0]$	$[1]$	$z + 2$	undefined
7.2.7.16a1.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 21 ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$F_7$ (as 14T4)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}]_{3}^{2}$	$[\frac{1}{3}]_{3}^{2}$	$[\ ]_{3}$	$[\ ]_{3}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 21 x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + 3$	undefined
7.2.7.16a2.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 42 ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$F_7$ (as 14T4)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}]_{3}^{2}$	$[\frac{1}{3}]_{3}^{2}$	$[\ ]_{3}$	$[\ ]_{3}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 42 x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + 6$	undefined
7.2.7.16a3.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 14 ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$(C_7:C_3) \times C_2$ (as 14T5)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}]_{3}^{2}$	$[\frac{1}{3}]_{3}^{2}$	$[\ ]_{3}$	$[\ ]_{3}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 14 x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + 2$	undefined
7.2.7.16a4.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 35 ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$F_7$ (as 14T4)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}]_{3}^{2}$	$[\frac{1}{3}]_{3}^{2}$	$[\ ]_{3}$	$[\ ]_{3}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 35 x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + 5$	undefined
7.2.7.16a5.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 7 ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$(C_7:C_3) \times C_2$ (as 14T5)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}]_{3}^{2}$	$[\frac{1}{3}]_{3}^{2}$	$[\ ]_{3}$	$[\ ]_{3}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 7 x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + 1$	undefined
7.2.7.16a6.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 28 ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$(C_7:C_3) \times C_2$ (as 14T5)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}]_{3}^{2}$	$[\frac{1}{3}]_{3}^{2}$	$[\ ]_{3}$	$[\ ]_{3}$	$2$	$t^{2} + 6 t + 3$	$x^{7} + 28 x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + 4$	undefined
7.2.7.16a7.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 21 x ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{4}$	$[\frac{4}{3}]^{2}_{3}$	$[\frac{1}{3}]^{2}_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + 21 t x^{2} + 7$	$[2, 0]$	$[2]$	$z^2 + (4 t + 3)$	undefined
7.2.7.16a8.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(21 x + 21\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{4}$	$[\frac{4}{3}]^{2}_{3}$	$[\frac{1}{3}]^{2}_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(28 t + 42\right) x^{2} + 7$	$[2, 0]$	$[2]$	$z^2 + (4 t + 6)$	undefined
7.2.7.16a9.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(21 x + 35\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7:F_7$ (as 14T14)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{2}$	$[\frac{4}{3}]_{3}$	$[\frac{1}{3}]_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(21 t + 35\right) x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + (4 t + 1)$	undefined
7.2.7.16a10.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(21 x + 7\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{4}$	$[\frac{4}{3}]^{2}_{3}$	$[\frac{1}{3}]^{2}_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(21 t + 7\right) x^{2} + 7$	$[2, 0]$	$[2]$	$z^2 + (4 t + 4)$	undefined
7.2.7.16a11.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 42 x ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{4}$	$[\frac{4}{3}]^{2}_{3}$	$[\frac{1}{3}]^{2}_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 42\right) x^{2} + 7$	$[2, 0]$	$[2]$	$z^2 + (t + 6)$	undefined
7.2.7.16a12.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(42 x + 21\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7:F_7$ (as 14T14)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{2}$	$[\frac{4}{3}]_{3}$	$[\frac{1}{3}]_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(42 t + 21\right) x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + (t + 2)$	undefined
7.2.7.16a13.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(42 x + 14\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{4}$	$[\frac{4}{3}]^{2}_{3}$	$[\frac{1}{3}]^{2}_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 7\right) x^{2} + 7$	$[2, 0]$	$[2]$	$z^2 + (t + 1)$	undefined
7.2.7.16a14.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(42 x + 35\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7:F_7$ (as 14T14)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{2}$	$[\frac{4}{3}]_{3}$	$[\frac{1}{3}]_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 28\right) x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + (t + 4)$	undefined
7.2.7.16a15.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(42 x + 28\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7:F_7$ (as 14T14)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{2}$	$[\frac{4}{3}]_{3}$	$[\frac{1}{3}]_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(7 t + 21\right) x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + (t + 3)$	undefined
7.2.7.16a16.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 14 x ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{4}$	$[\frac{4}{3}]^{2}_{3}$	$[\frac{1}{3}]^{2}_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + 14 t x^{2} + 7$	$[2, 0]$	$[2]$	$z^2 + (5 t + 2)$	undefined
7.2.7.16a17.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(14 x + 21\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{4}$	$[\frac{4}{3}]^{2}_{3}$	$[\frac{1}{3}]^{2}_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 21\right) x^{2} + 7$	$[2, 0]$	$[2]$	$z^2 + (5 t + 5)$	undefined
7.2.7.16a18.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(14 x + 14\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{4}$	$[\frac{4}{3}]^{2}_{3}$	$[\frac{1}{3}]^{2}_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 14\right) x^{2} + 7$	$[2, 0]$	$[2]$	$z^2 + (5 t + 4)$	undefined
7.2.7.16a19.1	$14$	$( x^{2} + 6 x + 3 )^{7} + \left(14 x + 7\right) ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7:F_7$ (as 14T14)	$2$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{2}$	$[\frac{4}{3}]_{3}$	$[\frac{1}{3}]_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 7\right) x^{2} + 7$	$[2, 0]$	$[1]$	$z^2 + (5 t + 3)$	undefined
7.2.7.16a20.1	$14$	$( x^{2} + 6 x + 3 )^{7} + 35 x ( x^{2} + 6 x + 3 )^{2} + 7$	$7$	$2$	$7$	$16$	$C_7^2:C_{12}$ (as 14T23)	$4$	$3$	$[\frac{4}{3}]$	$[\frac{1}{3}]$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{4}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{4}$	$[\frac{4}{3}]^{2}_{3}$	$[\frac{1}{3}]^{2}_{3}$	$1$	$t^{2} + 6 t + 3$	$x^{7} + \left(14 t + 35\right) x^{2} + 7$	$[2, 0]$	$[2]$	$z^2 + (2 t + 5)$	undefined

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