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Properties

Label 32.0.50247333250...5889.1
Degree $32$
Signature $[0, 16]$
Discriminant $17^{30}\cdot 23^{16}$
Root discriminant $68.30$
Ramified primes $17, 23$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2821109907456, -470184984576, -391820820480, 143667634176, 41358864384, -30837749760, -1753519104, 5431878144, -613059840, -803136384, 236032704, 94517280, -55091664, -6570936, 10277100, -617694, -1609901, -102949, 285475, -30421, -42509, 12155, 5059, -2869, -365, 539, -29, -85, 19, 11, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 5*x^30 + 11*x^29 + 19*x^28 - 85*x^27 - 29*x^26 + 539*x^25 - 365*x^24 - 2869*x^23 + 5059*x^22 + 12155*x^21 - 42509*x^20 - 30421*x^19 + 285475*x^18 - 102949*x^17 - 1609901*x^16 - 617694*x^15 + 10277100*x^14 - 6570936*x^13 - 55091664*x^12 + 94517280*x^11 + 236032704*x^10 - 803136384*x^9 - 613059840*x^8 + 5431878144*x^7 - 1753519104*x^6 - 30837749760*x^5 + 41358864384*x^4 + 143667634176*x^3 - 391820820480*x^2 - 470184984576*x + 2821109907456)
 
gp: K = bnfinit(x^32 - x^31 - 5*x^30 + 11*x^29 + 19*x^28 - 85*x^27 - 29*x^26 + 539*x^25 - 365*x^24 - 2869*x^23 + 5059*x^22 + 12155*x^21 - 42509*x^20 - 30421*x^19 + 285475*x^18 - 102949*x^17 - 1609901*x^16 - 617694*x^15 + 10277100*x^14 - 6570936*x^13 - 55091664*x^12 + 94517280*x^11 + 236032704*x^10 - 803136384*x^9 - 613059840*x^8 + 5431878144*x^7 - 1753519104*x^6 - 30837749760*x^5 + 41358864384*x^4 + 143667634176*x^3 - 391820820480*x^2 - 470184984576*x + 2821109907456, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 5 x^{30} + 11 x^{29} + 19 x^{28} - 85 x^{27} - 29 x^{26} + 539 x^{25} - 365 x^{24} - 2869 x^{23} + 5059 x^{22} + 12155 x^{21} - 42509 x^{20} - 30421 x^{19} + 285475 x^{18} - 102949 x^{17} - 1609901 x^{16} - 617694 x^{15} + 10277100 x^{14} - 6570936 x^{13} - 55091664 x^{12} + 94517280 x^{11} + 236032704 x^{10} - 803136384 x^{9} - 613059840 x^{8} + 5431878144 x^{7} - 1753519104 x^{6} - 30837749760 x^{5} + 41358864384 x^{4} + 143667634176 x^{3} - 391820820480 x^{2} - 470184984576 x + 2821109907456 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(50247333250656904702494452824050108123783217432758438465889=17^{30}\cdot 23^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.30$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $17, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(391=17\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{391}(1,·)$, $\chi_{391}(390,·)$, $\chi_{391}(137,·)$, $\chi_{391}(139,·)$, $\chi_{391}(275,·)$, $\chi_{391}(277,·)$, $\chi_{391}(22,·)$, $\chi_{391}(24,·)$, $\chi_{391}(160,·)$, $\chi_{391}(162,·)$, $\chi_{391}(298,·)$, $\chi_{391}(300,·)$, $\chi_{391}(45,·)$, $\chi_{391}(47,·)$, $\chi_{391}(183,·)$, $\chi_{391}(185,·)$, $\chi_{391}(321,·)$, $\chi_{391}(70,·)$, $\chi_{391}(206,·)$, $\chi_{391}(208,·)$, $\chi_{391}(344,·)$, $\chi_{391}(346,·)$, $\chi_{391}(91,·)$, $\chi_{391}(93,·)$, $\chi_{391}(229,·)$, $\chi_{391}(231,·)$, $\chi_{391}(367,·)$, $\chi_{391}(369,·)$, $\chi_{391}(114,·)$, $\chi_{391}(116,·)$, $\chi_{391}(252,·)$, $\chi_{391}(254,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{9659406} a^{17} + \frac{1}{6} a^{16} - \frac{1}{6} a^{15} + \frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{102949}{1609901}$, $\frac{1}{57956436} a^{18} - \frac{1}{57956436} a^{17} - \frac{13}{36} a^{16} + \frac{7}{36} a^{15} - \frac{1}{36} a^{14} - \frac{5}{36} a^{13} + \frac{11}{36} a^{12} - \frac{17}{36} a^{11} - \frac{13}{36} a^{10} + \frac{7}{36} a^{9} - \frac{1}{36} a^{8} - \frac{5}{36} a^{7} + \frac{11}{36} a^{6} - \frac{17}{36} a^{5} - \frac{13}{36} a^{4} + \frac{7}{36} a^{3} - \frac{1}{36} a^{2} - \frac{102949}{9659406} a + \frac{285475}{1609901}$, $\frac{1}{347738616} a^{19} - \frac{1}{347738616} a^{18} - \frac{5}{347738616} a^{17} - \frac{65}{216} a^{16} - \frac{73}{216} a^{15} + \frac{31}{216} a^{14} - \frac{25}{216} a^{13} + \frac{55}{216} a^{12} + \frac{95}{216} a^{11} + \frac{7}{216} a^{10} + \frac{71}{216} a^{9} + \frac{103}{216} a^{8} - \frac{97}{216} a^{7} - \frac{89}{216} a^{6} + \frac{23}{216} a^{5} + \frac{79}{216} a^{4} - \frac{1}{216} a^{3} - \frac{102949}{57956436} a^{2} + \frac{285475}{9659406} a - \frac{30421}{1609901}$, $\frac{1}{2086431696} a^{20} - \frac{1}{2086431696} a^{19} - \frac{5}{2086431696} a^{18} + \frac{11}{2086431696} a^{17} - \frac{289}{1296} a^{16} - \frac{617}{1296} a^{15} - \frac{241}{1296} a^{14} + \frac{55}{1296} a^{13} + \frac{95}{1296} a^{12} - \frac{425}{1296} a^{11} - \frac{145}{1296} a^{10} + \frac{103}{1296} a^{9} - \frac{529}{1296} a^{8} - \frac{89}{1296} a^{7} - \frac{625}{1296} a^{6} - \frac{137}{1296} a^{5} - \frac{1}{1296} a^{4} - \frac{102949}{347738616} a^{3} + \frac{285475}{57956436} a^{2} - \frac{30421}{9659406} a - \frac{42509}{1609901}$, $\frac{1}{12518590176} a^{21} - \frac{1}{12518590176} a^{20} - \frac{5}{12518590176} a^{19} + \frac{11}{12518590176} a^{18} + \frac{19}{12518590176} a^{17} - \frac{3209}{7776} a^{16} - \frac{2833}{7776} a^{15} - \frac{1241}{7776} a^{14} + \frac{2687}{7776} a^{13} - \frac{3017}{7776} a^{12} + \frac{2447}{7776} a^{11} + \frac{103}{7776} a^{10} + \frac{767}{7776} a^{9} - \frac{1385}{7776} a^{8} - \frac{3217}{7776} a^{7} + \frac{3751}{7776} a^{6} - \frac{1}{7776} a^{5} - \frac{102949}{2086431696} a^{4} + \frac{285475}{347738616} a^{3} - \frac{30421}{57956436} a^{2} - \frac{42509}{9659406} a + \frac{12155}{1609901}$, $\frac{1}{75111541056} a^{22} - \frac{1}{75111541056} a^{21} - \frac{5}{75111541056} a^{20} + \frac{11}{75111541056} a^{19} + \frac{19}{75111541056} a^{18} - \frac{85}{75111541056} a^{17} + \frac{4943}{46656} a^{16} + \frac{14311}{46656} a^{15} + \frac{2687}{46656} a^{14} + \frac{4759}{46656} a^{13} - \frac{20881}{46656} a^{12} - \frac{7673}{46656} a^{11} - \frac{7009}{46656} a^{10} + \frac{6391}{46656} a^{9} - \frac{10993}{46656} a^{8} + \frac{19303}{46656} a^{7} - \frac{1}{46656} a^{6} - \frac{102949}{12518590176} a^{5} + \frac{285475}{2086431696} a^{4} - \frac{30421}{347738616} a^{3} - \frac{42509}{57956436} a^{2} + \frac{12155}{9659406} a + \frac{5059}{1609901}$, $\frac{1}{450669246336} a^{23} - \frac{1}{450669246336} a^{22} - \frac{5}{450669246336} a^{21} + \frac{11}{450669246336} a^{20} + \frac{19}{450669246336} a^{19} - \frac{85}{450669246336} a^{18} - \frac{29}{450669246336} a^{17} + \frac{14311}{279936} a^{16} - \frac{43969}{279936} a^{15} - \frac{41897}{279936} a^{14} + \frac{25775}{279936} a^{13} - \frac{54329}{279936} a^{12} - \frac{100321}{279936} a^{11} - \frac{133577}{279936} a^{10} - \frac{104305}{279936} a^{9} + \frac{65959}{279936} a^{8} - \frac{1}{279936} a^{7} - \frac{102949}{75111541056} a^{6} + \frac{285475}{12518590176} a^{5} - \frac{30421}{2086431696} a^{4} - \frac{42509}{347738616} a^{3} + \frac{12155}{57956436} a^{2} + \frac{5059}{9659406} a - \frac{2869}{1609901}$, $\frac{1}{2704015478016} a^{24} - \frac{1}{2704015478016} a^{23} - \frac{5}{2704015478016} a^{22} + \frac{11}{2704015478016} a^{21} + \frac{19}{2704015478016} a^{20} - \frac{85}{2704015478016} a^{19} - \frac{29}{2704015478016} a^{18} + \frac{539}{2704015478016} a^{17} - \frac{43969}{1679616} a^{16} - \frac{41897}{1679616} a^{15} + \frac{305711}{1679616} a^{14} - \frac{54329}{1679616} a^{13} - \frac{100321}{1679616} a^{12} + \frac{426295}{1679616} a^{11} + \frac{175631}{1679616} a^{10} + \frac{625831}{1679616} a^{9} - \frac{1}{1679616} a^{8} - \frac{102949}{450669246336} a^{7} + \frac{285475}{75111541056} a^{6} - \frac{30421}{12518590176} a^{5} - \frac{42509}{2086431696} a^{4} + \frac{12155}{347738616} a^{3} + \frac{5059}{57956436} a^{2} - \frac{2869}{9659406} a - \frac{365}{1609901}$, $\frac{1}{16224092868096} a^{25} - \frac{1}{16224092868096} a^{24} - \frac{5}{16224092868096} a^{23} + \frac{11}{16224092868096} a^{22} + \frac{19}{16224092868096} a^{21} - \frac{85}{16224092868096} a^{20} - \frac{29}{16224092868096} a^{19} + \frac{539}{16224092868096} a^{18} - \frac{365}{16224092868096} a^{17} - \frac{41897}{10077696} a^{16} + \frac{305711}{10077696} a^{15} - \frac{54329}{10077696} a^{14} - \frac{1779937}{10077696} a^{13} + \frac{2105911}{10077696} a^{12} - \frac{1503985}{10077696} a^{11} - \frac{1053785}{10077696} a^{10} - \frac{1}{10077696} a^{9} - \frac{102949}{2704015478016} a^{8} + \frac{285475}{450669246336} a^{7} - \frac{30421}{75111541056} a^{6} - \frac{42509}{12518590176} a^{5} + \frac{12155}{2086431696} a^{4} + \frac{5059}{347738616} a^{3} - \frac{2869}{57956436} a^{2} - \frac{365}{9659406} a + \frac{539}{1609901}$, $\frac{1}{97344557208576} a^{26} - \frac{1}{97344557208576} a^{25} - \frac{5}{97344557208576} a^{24} + \frac{11}{97344557208576} a^{23} + \frac{19}{97344557208576} a^{22} - \frac{85}{97344557208576} a^{21} - \frac{29}{97344557208576} a^{20} + \frac{539}{97344557208576} a^{19} - \frac{365}{97344557208576} a^{18} - \frac{2869}{97344557208576} a^{17} - \frac{29927377}{60466176} a^{16} + \frac{30178759}{60466176} a^{15} + \frac{28453151}{60466176} a^{14} - \frac{28127177}{60466176} a^{13} - \frac{21659377}{60466176} a^{12} + \frac{9023911}{60466176} a^{11} - \frac{1}{60466176} a^{10} - \frac{102949}{16224092868096} a^{9} + \frac{285475}{2704015478016} a^{8} - \frac{30421}{450669246336} a^{7} - \frac{42509}{75111541056} a^{6} + \frac{12155}{12518590176} a^{5} + \frac{5059}{2086431696} a^{4} - \frac{2869}{347738616} a^{3} - \frac{365}{57956436} a^{2} + \frac{539}{9659406} a - \frac{29}{1609901}$, $\frac{1}{584067343251456} a^{27} - \frac{1}{584067343251456} a^{26} - \frac{5}{584067343251456} a^{25} + \frac{11}{584067343251456} a^{24} + \frac{19}{584067343251456} a^{23} - \frac{85}{584067343251456} a^{22} - \frac{29}{584067343251456} a^{21} + \frac{539}{584067343251456} a^{20} - \frac{365}{584067343251456} a^{19} - \frac{2869}{584067343251456} a^{18} + \frac{5059}{584067343251456} a^{17} + \frac{151111111}{362797056} a^{16} + \frac{28453151}{362797056} a^{15} + \frac{153271351}{362797056} a^{14} + \frac{38806799}{362797056} a^{13} + \frac{129956263}{362797056} a^{12} - \frac{1}{362797056} a^{11} - \frac{102949}{97344557208576} a^{10} + \frac{285475}{16224092868096} a^{9} - \frac{30421}{2704015478016} a^{8} - \frac{42509}{450669246336} a^{7} + \frac{12155}{75111541056} a^{6} + \frac{5059}{12518590176} a^{5} - \frac{2869}{2086431696} a^{4} - \frac{365}{347738616} a^{3} + \frac{539}{57956436} a^{2} - \frac{29}{9659406} a - \frac{85}{1609901}$, $\frac{1}{3504404059508736} a^{28} - \frac{1}{3504404059508736} a^{27} - \frac{5}{3504404059508736} a^{26} + \frac{11}{3504404059508736} a^{25} + \frac{19}{3504404059508736} a^{24} - \frac{85}{3504404059508736} a^{23} - \frac{29}{3504404059508736} a^{22} + \frac{539}{3504404059508736} a^{21} - \frac{365}{3504404059508736} a^{20} - \frac{2869}{3504404059508736} a^{19} + \frac{5059}{3504404059508736} a^{18} + \frac{12155}{3504404059508736} a^{17} + \frac{754047263}{2176782336} a^{16} + \frac{516068407}{2176782336} a^{15} - \frac{686787313}{2176782336} a^{14} - \frac{232840793}{2176782336} a^{13} - \frac{1}{2176782336} a^{12} - \frac{102949}{584067343251456} a^{11} + \frac{285475}{97344557208576} a^{10} - \frac{30421}{16224092868096} a^{9} - \frac{42509}{2704015478016} a^{8} + \frac{12155}{450669246336} a^{7} + \frac{5059}{75111541056} a^{6} - \frac{2869}{12518590176} a^{5} - \frac{365}{2086431696} a^{4} + \frac{539}{347738616} a^{3} - \frac{29}{57956436} a^{2} - \frac{85}{9659406} a + \frac{19}{1609901}$, $\frac{1}{21026424357052416} a^{29} - \frac{1}{21026424357052416} a^{28} - \frac{5}{21026424357052416} a^{27} + \frac{11}{21026424357052416} a^{26} + \frac{19}{21026424357052416} a^{25} - \frac{85}{21026424357052416} a^{24} - \frac{29}{21026424357052416} a^{23} + \frac{539}{21026424357052416} a^{22} - \frac{365}{21026424357052416} a^{21} - \frac{2869}{21026424357052416} a^{20} + \frac{5059}{21026424357052416} a^{19} + \frac{12155}{21026424357052416} a^{18} - \frac{42509}{21026424357052416} a^{17} + \frac{2692850743}{13060694016} a^{16} + \frac{5843559695}{13060694016} a^{15} + \frac{4120723879}{13060694016} a^{14} - \frac{1}{13060694016} a^{13} - \frac{102949}{3504404059508736} a^{12} + \frac{285475}{584067343251456} a^{11} - \frac{30421}{97344557208576} a^{10} - \frac{42509}{16224092868096} a^{9} + \frac{12155}{2704015478016} a^{8} + \frac{5059}{450669246336} a^{7} - \frac{2869}{75111541056} a^{6} - \frac{365}{12518590176} a^{5} + \frac{539}{2086431696} a^{4} - \frac{29}{347738616} a^{3} - \frac{85}{57956436} a^{2} + \frac{19}{9659406} a + \frac{11}{1609901}$, $\frac{1}{126158546142314496} a^{30} - \frac{1}{126158546142314496} a^{29} - \frac{5}{126158546142314496} a^{28} + \frac{11}{126158546142314496} a^{27} + \frac{19}{126158546142314496} a^{26} - \frac{85}{126158546142314496} a^{25} - \frac{29}{126158546142314496} a^{24} + \frac{539}{126158546142314496} a^{23} - \frac{365}{126158546142314496} a^{22} - \frac{2869}{126158546142314496} a^{21} + \frac{5059}{126158546142314496} a^{20} + \frac{12155}{126158546142314496} a^{19} - \frac{42509}{126158546142314496} a^{18} - \frac{30421}{126158546142314496} a^{17} + \frac{18904253711}{78364164096} a^{16} - \frac{35061358169}{78364164096} a^{15} - \frac{1}{78364164096} a^{14} - \frac{102949}{21026424357052416} a^{13} + \frac{285475}{3504404059508736} a^{12} - \frac{30421}{584067343251456} a^{11} - \frac{42509}{97344557208576} a^{10} + \frac{12155}{16224092868096} a^{9} + \frac{5059}{2704015478016} a^{8} - \frac{2869}{450669246336} a^{7} - \frac{365}{75111541056} a^{6} + \frac{539}{12518590176} a^{5} - \frac{29}{2086431696} a^{4} - \frac{85}{347738616} a^{3} + \frac{19}{57956436} a^{2} + \frac{11}{9659406} a - \frac{5}{1609901}$, $\frac{1}{756951276853886976} a^{31} - \frac{1}{756951276853886976} a^{30} - \frac{5}{756951276853886976} a^{29} + \frac{11}{756951276853886976} a^{28} + \frac{19}{756951276853886976} a^{27} - \frac{85}{756951276853886976} a^{26} - \frac{29}{756951276853886976} a^{25} + \frac{539}{756951276853886976} a^{24} - \frac{365}{756951276853886976} a^{23} - \frac{2869}{756951276853886976} a^{22} + \frac{5059}{756951276853886976} a^{21} + \frac{12155}{756951276853886976} a^{20} - \frac{42509}{756951276853886976} a^{19} - \frac{30421}{756951276853886976} a^{18} + \frac{285475}{756951276853886976} a^{17} - \frac{113425522265}{470184984576} a^{16} - \frac{1}{470184984576} a^{15} - \frac{102949}{126158546142314496} a^{14} + \frac{285475}{21026424357052416} a^{13} - \frac{30421}{3504404059508736} a^{12} - \frac{42509}{584067343251456} a^{11} + \frac{12155}{97344557208576} a^{10} + \frac{5059}{16224092868096} a^{9} - \frac{2869}{2704015478016} a^{8} - \frac{365}{450669246336} a^{7} + \frac{539}{75111541056} a^{6} - \frac{29}{12518590176} a^{5} - \frac{85}{2086431696} a^{4} + \frac{19}{347738616} a^{3} + \frac{11}{57956436} a^{2} - \frac{5}{9659406} a - \frac{1}{1609901}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{42509}{21026424357052416} a^{30} + \frac{22717244075}{21026424357052416} a^{13} \) (order $34$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-391}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{17}, \sqrt{-23})\), 4.4.4913.1, 4.0.2598977.1, 8.0.6754681446529.1, \(\Q(\zeta_{17})^+\), 8.0.114829584590993.1, 16.0.13185833497340017023096726049.1, 16.16.224159169454780289392644342833.1, \(\Q(\zeta_{17})\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ R $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed
23Data not computed