Properties

Label 30.0.607...847.1
Degree $30$
Signature $[0, 15]$
Discriminant $-6.072\times 10^{48}$
Root discriminant $42.28$
Ramified primes $17, 127$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 6*x^29 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849)
 
gp: K = bnfinit(x^30 - 6*x^29 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1849, -19350, 103267, -363451, 950326, -1976201, 3416179, -5050948, 6524456, -7475970, 7698034, -7195888, 6168845, -4892313, 3620042, -2519432, 1659175, -1037914, 617591, -348333, 185209, -92148, 42428, -17890, 6840, -2325, 694, -178, 38, -6, 1]);
 

\( x^{30} - 6 x^{29} + 38 x^{28} - 178 x^{27} + 694 x^{26} - 2325 x^{25} + 6840 x^{24} - 17890 x^{23} + 42428 x^{22} - 92148 x^{21} + 185209 x^{20} - 348333 x^{19} + 617591 x^{18} - 1037914 x^{17} + 1659175 x^{16} - 2519432 x^{15} + 3620042 x^{14} - 4892313 x^{13} + 6168845 x^{12} - 7195888 x^{11} + 7698034 x^{10} - 7475970 x^{9} + 6524456 x^{8} - 5050948 x^{7} + 3416179 x^{6} - 1976201 x^{5} + 950326 x^{4} - 363451 x^{3} + 103267 x^{2} - 19350 x + 1849 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-6072125144349637560639895035214067242224098130847\)\(\medspace = -\,17^{14}\cdot 127^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $42.28$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $17, 127$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17} a^{11} - \frac{1}{17} a^{10} + \frac{1}{17} a^{9} + \frac{6}{17} a^{8} + \frac{8}{17} a^{7} - \frac{8}{17} a^{6} - \frac{7}{17} a^{4} - \frac{1}{17} a^{3} + \frac{2}{17} a^{2} - \frac{7}{17} a - \frac{7}{17}$, $\frac{1}{17} a^{12} + \frac{7}{17} a^{9} - \frac{3}{17} a^{8} - \frac{8}{17} a^{6} - \frac{7}{17} a^{5} - \frac{8}{17} a^{4} + \frac{1}{17} a^{3} - \frac{5}{17} a^{2} + \frac{3}{17} a - \frac{7}{17}$, $\frac{1}{17} a^{13} + \frac{7}{17} a^{10} - \frac{3}{17} a^{9} - \frac{8}{17} a^{7} - \frac{7}{17} a^{6} - \frac{8}{17} a^{5} + \frac{1}{17} a^{4} - \frac{5}{17} a^{3} + \frac{3}{17} a^{2} - \frac{7}{17} a$, $\frac{1}{17} a^{14} + \frac{4}{17} a^{10} - \frac{7}{17} a^{9} + \frac{1}{17} a^{8} + \frac{5}{17} a^{7} - \frac{3}{17} a^{6} + \frac{1}{17} a^{5} - \frac{7}{17} a^{4} - \frac{7}{17} a^{3} - \frac{4}{17} a^{2} - \frac{2}{17} a - \frac{2}{17}$, $\frac{1}{17} a^{15} - \frac{3}{17} a^{10} - \frac{3}{17} a^{9} - \frac{2}{17} a^{8} - \frac{1}{17} a^{7} - \frac{1}{17} a^{6} - \frac{7}{17} a^{5} + \frac{4}{17} a^{4} + \frac{7}{17} a^{2} - \frac{8}{17} a - \frac{6}{17}$, $\frac{1}{17} a^{16} - \frac{6}{17} a^{10} + \frac{1}{17} a^{9} + \frac{6}{17} a^{7} + \frac{3}{17} a^{6} + \frac{4}{17} a^{5} - \frac{4}{17} a^{4} + \frac{4}{17} a^{3} - \frac{2}{17} a^{2} + \frac{7}{17} a - \frac{4}{17}$, $\frac{1}{17} a^{17} - \frac{5}{17} a^{10} + \frac{6}{17} a^{9} + \frac{8}{17} a^{8} + \frac{7}{17} a^{6} - \frac{4}{17} a^{5} - \frac{4}{17} a^{4} - \frac{8}{17} a^{3} + \frac{2}{17} a^{2} + \frac{5}{17} a - \frac{8}{17}$, $\frac{1}{17} a^{18} + \frac{1}{17} a^{10} - \frac{4}{17} a^{9} - \frac{4}{17} a^{8} - \frac{4}{17} a^{7} + \frac{7}{17} a^{6} - \frac{4}{17} a^{5} + \frac{8}{17} a^{4} - \frac{3}{17} a^{3} - \frac{2}{17} a^{2} + \frac{8}{17} a - \frac{1}{17}$, $\frac{1}{17} a^{19} - \frac{3}{17} a^{10} - \frac{5}{17} a^{9} + \frac{7}{17} a^{8} - \frac{1}{17} a^{7} + \frac{4}{17} a^{6} + \frac{8}{17} a^{5} + \frac{4}{17} a^{4} - \frac{1}{17} a^{3} + \frac{6}{17} a^{2} + \frac{6}{17} a + \frac{7}{17}$, $\frac{1}{17} a^{20} - \frac{8}{17} a^{10} - \frac{7}{17} a^{9} - \frac{6}{17} a^{7} + \frac{1}{17} a^{6} + \frac{4}{17} a^{5} - \frac{5}{17} a^{4} + \frac{3}{17} a^{3} - \frac{5}{17} a^{2} + \frac{3}{17} a - \frac{4}{17}$, $\frac{1}{187} a^{21} + \frac{2}{187} a^{20} - \frac{3}{187} a^{19} - \frac{3}{187} a^{18} - \frac{3}{187} a^{17} - \frac{3}{187} a^{16} + \frac{3}{187} a^{15} + \frac{1}{187} a^{13} + \frac{1}{187} a^{12} + \frac{2}{187} a^{11} + \frac{4}{187} a^{10} - \frac{54}{187} a^{9} + \frac{63}{187} a^{8} + \frac{38}{187} a^{7} - \frac{19}{187} a^{6} + \frac{23}{187} a^{5} + \frac{69}{187} a^{4} + \frac{28}{187} a^{3} + \frac{37}{187} a^{2} - \frac{4}{187} a + \frac{4}{11}$, $\frac{1}{3179} a^{22} - \frac{2}{3179} a^{21} + \frac{8}{289} a^{20} - \frac{24}{3179} a^{19} - \frac{46}{3179} a^{18} + \frac{31}{3179} a^{17} - \frac{3}{187} a^{16} + \frac{32}{3179} a^{15} - \frac{54}{3179} a^{14} + \frac{5}{187} a^{13} - \frac{57}{3179} a^{12} + \frac{84}{3179} a^{11} + \frac{392}{3179} a^{10} + \frac{1412}{3179} a^{9} - \frac{808}{3179} a^{8} - \frac{666}{3179} a^{7} - \frac{8}{17} a^{6} - \frac{892}{3179} a^{5} - \frac{105}{3179} a^{4} - \frac{966}{3179} a^{3} + \frac{310}{3179} a^{2} - \frac{1500}{3179} a - \frac{580}{3179}$, $\frac{1}{3179} a^{23} - \frac{1}{3179} a^{21} - \frac{18}{3179} a^{20} - \frac{26}{3179} a^{19} + \frac{7}{3179} a^{18} + \frac{79}{3179} a^{17} - \frac{2}{3179} a^{16} - \frac{58}{3179} a^{15} - \frac{23}{3179} a^{14} + \frac{28}{3179} a^{13} + \frac{72}{3179} a^{12} + \frac{16}{3179} a^{11} + \frac{921}{3179} a^{10} + \frac{996}{3179} a^{9} - \frac{157}{3179} a^{8} + \frac{113}{3179} a^{7} + \frac{1471}{3179} a^{6} - \frac{852}{3179} a^{5} + \frac{439}{3179} a^{4} + \frac{1234}{3179} a^{3} + \frac{1211}{3179} a^{2} - \frac{61}{3179} a + \frac{727}{3179}$, $\frac{1}{3179} a^{24} - \frac{3}{3179} a^{21} - \frac{91}{3179} a^{20} - \frac{4}{187} a^{19} - \frac{18}{3179} a^{18} - \frac{2}{289} a^{17} + \frac{27}{3179} a^{16} + \frac{60}{3179} a^{15} - \frac{26}{3179} a^{14} - \frac{13}{3179} a^{13} - \frac{24}{3179} a^{12} - \frac{83}{3179} a^{11} - \frac{1536}{3179} a^{10} + \frac{1272}{3179} a^{9} + \frac{2}{3179} a^{8} - \frac{606}{3179} a^{7} - \frac{1549}{3179} a^{6} + \frac{1434}{3179} a^{5} + \frac{619}{3179} a^{4} - \frac{214}{3179} a^{3} - \frac{1366}{3179} a^{2} - \frac{467}{3179} a - \frac{1107}{3179}$, $\frac{1}{3179} a^{25} + \frac{5}{3179} a^{21} + \frac{26}{3179} a^{20} - \frac{2}{289} a^{19} - \frac{92}{3179} a^{18} + \frac{1}{3179} a^{17} - \frac{25}{3179} a^{16} + \frac{2}{3179} a^{15} + \frac{12}{3179} a^{14} - \frac{41}{3179} a^{13} + \frac{35}{3179} a^{12} + \frac{42}{3179} a^{11} + \frac{3}{187} a^{10} - \frac{335}{3179} a^{9} + \frac{404}{3179} a^{8} + \frac{329}{3179} a^{7} - \frac{504}{3179} a^{6} - \frac{49}{187} a^{5} + \frac{1273}{3179} a^{4} - \frac{111}{289} a^{3} + \frac{871}{3179} a^{2} - \frac{405}{3179} a - \frac{1349}{3179}$, $\frac{1}{3179} a^{26} + \frac{2}{3179} a^{21} + \frac{31}{3179} a^{20} - \frac{57}{3179} a^{19} - \frac{41}{3179} a^{18} - \frac{78}{3179} a^{17} - \frac{15}{3179} a^{16} - \frac{63}{3179} a^{15} + \frac{42}{3179} a^{14} - \frac{50}{3179} a^{13} - \frac{81}{3179} a^{12} - \frac{63}{3179} a^{11} - \frac{2}{17} a^{10} + \frac{229}{3179} a^{9} - \frac{67}{187} a^{8} - \frac{336}{3179} a^{7} + \frac{6}{17} a^{6} + \frac{650}{3179} a^{5} + \frac{1072}{3179} a^{4} - \frac{1422}{3179} a^{3} + \frac{9}{187} a^{2} + \frac{116}{3179} a + \frac{1336}{3179}$, $\frac{1}{2323849} a^{27} + \frac{256}{2323849} a^{26} + \frac{15}{211259} a^{25} - \frac{12}{211259} a^{24} + \frac{23}{211259} a^{23} + \frac{2}{2323849} a^{22} + \frac{2174}{2323849} a^{21} - \frac{57213}{2323849} a^{20} + \frac{51897}{2323849} a^{19} - \frac{783}{136697} a^{18} + \frac{51574}{2323849} a^{17} - \frac{17266}{2323849} a^{16} - \frac{63476}{2323849} a^{15} - \frac{60952}{2323849} a^{14} - \frac{214}{136697} a^{13} - \frac{14482}{2323849} a^{12} + \frac{35743}{2323849} a^{11} - \frac{316460}{2323849} a^{10} - \frac{341086}{2323849} a^{9} - \frac{80904}{211259} a^{8} - \frac{781905}{2323849} a^{7} + \frac{1113771}{2323849} a^{6} + \frac{104553}{211259} a^{5} + \frac{8930}{136697} a^{4} - \frac{504408}{2323849} a^{3} + \frac{929658}{2323849} a^{2} - \frac{90703}{211259} a - \frac{15322}{54043}$, $\frac{1}{8744643787} a^{28} - \frac{1385}{8744643787} a^{27} - \frac{48476}{794967617} a^{26} + \frac{319751}{8744643787} a^{25} - \frac{1272913}{8744643787} a^{24} - \frac{953918}{8744643787} a^{23} + \frac{1191884}{8744643787} a^{22} - \frac{23254290}{8744643787} a^{21} - \frac{10412375}{514390811} a^{20} - \frac{252428357}{8744643787} a^{19} + \frac{103882423}{8744643787} a^{18} + \frac{222523310}{8744643787} a^{17} + \frac{47425154}{8744643787} a^{16} + \frac{84369088}{8744643787} a^{15} + \frac{233676827}{8744643787} a^{14} + \frac{133316875}{8744643787} a^{13} + \frac{208612856}{8744643787} a^{12} + \frac{136757451}{8744643787} a^{11} + \frac{1724727227}{8744643787} a^{10} + \frac{72698176}{8744643787} a^{9} + \frac{2401216568}{8744643787} a^{8} + \frac{666368442}{8744643787} a^{7} + \frac{562403001}{8744643787} a^{6} - \frac{2639366288}{8744643787} a^{5} + \frac{1977451678}{8744643787} a^{4} - \frac{580022060}{8744643787} a^{3} + \frac{1786372273}{8744643787} a^{2} - \frac{4247656330}{8744643787} a - \frac{2504990}{203363809}$, $\frac{1}{10869592227241} a^{29} + \frac{394}{10869592227241} a^{28} - \frac{927501}{10869592227241} a^{27} - \frac{294692808}{10869592227241} a^{26} - \frac{17385894}{988144747931} a^{25} + \frac{1040069708}{10869592227241} a^{24} - \frac{919137512}{10869592227241} a^{23} - \frac{1180401683}{10869592227241} a^{22} - \frac{7492882953}{10869592227241} a^{21} + \frac{111183350176}{10869592227241} a^{20} - \frac{1430506986}{96191081657} a^{19} - \frac{2376810727}{153092848271} a^{18} - \frac{249737690075}{10869592227241} a^{17} + \frac{115215844176}{10869592227241} a^{16} - \frac{59682595032}{10869592227241} a^{15} + \frac{319394965449}{10869592227241} a^{14} + \frac{290548062479}{10869592227241} a^{13} - \frac{69194759450}{10869592227241} a^{12} + \frac{14786857864}{988144747931} a^{11} + \frac{885140395022}{10869592227241} a^{10} - \frac{451544042711}{10869592227241} a^{9} + \frac{2228595007301}{10869592227241} a^{8} - \frac{3855946038863}{10869592227241} a^{7} + \frac{3177923235990}{10869592227241} a^{6} + \frac{4707563738929}{10869592227241} a^{5} + \frac{82060550320}{252781214587} a^{4} + \frac{4269053233906}{10869592227241} a^{3} + \frac{920063676781}{10869592227241} a^{2} - \frac{3206359537761}{10869592227241} a + \frac{55691219114}{252781214587}$

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

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 588337897727.5111 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 588337897727.5111 \cdot 5}{2\sqrt{6072125144349637560639895035214067242224098130847}}\approx 0.560524373000263$ (assuming GRH)

Galois group

$D_{30}$ (as 30T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{-127}) \), 3.1.2159.1, 5.1.4661281.1, 6.0.591982687.1, 10.0.2759397651242047.1, 15.1.218659573334046061397519.1

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

Sibling fields

Degree 30 sibling: 30.2.812804153180660145912426894477473567856769041137.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15^{2}$ $30$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$ R $15^{2}$ $30$ $30$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{15}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
$127$127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.17.2t1.a.a$1$ $ 17 $ \(\Q(\sqrt{17}) \) $C_2$ (as 2T1) $1$ $1$
1.2159.2t1.a.a$1$ $ 17 \cdot 127 $ \(\Q(\sqrt{-2159}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.127.2t1.a.a$1$ $ 127 $ \(\Q(\sqrt{-127}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2159.6t3.b.a$2$ $ 17 \cdot 127 $ 6.2.79241777.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.2159.3t2.a.a$2$ $ 17 \cdot 127 $ 3.1.2159.1 $S_3$ (as 3T2) $1$ $0$
* 2.2159.5t2.a.a$2$ $ 17 \cdot 127 $ 5.1.4661281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2159.5t2.a.b$2$ $ 17 \cdot 127 $ 5.1.4661281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2159.10t3.a.a$2$ $ 17 \cdot 127 $ 10.2.369368189536337.2 $D_{10}$ (as 10T3) $1$ $0$
* 2.2159.10t3.a.b$2$ $ 17 \cdot 127 $ 10.2.369368189536337.2 $D_{10}$ (as 10T3) $1$ $0$
* 2.2159.15t2.a.c$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.a.d$2$ $ 17 \cdot 127 $ 30.0.6072125144349637560639895035214067242224098130847.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2159.15t2.a.a$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.a.b$2$ $ 17 \cdot 127 $ 30.0.6072125144349637560639895035214067242224098130847.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2159.15t2.a.d$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.a.a$2$ $ 17 \cdot 127 $ 30.0.6072125144349637560639895035214067242224098130847.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2159.15t2.a.b$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.a.c$2$ $ 17 \cdot 127 $ 30.0.6072125144349637560639895035214067242224098130847.1 $D_{30}$ (as 30T14) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.