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Properties

Label 20.0.386...125.1
Degree $20$
Signature $(0, 10)$
Discriminant $3.866\times 10^{26}$
Root discriminant \(21.35\)
Ramified primes $5,103$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4\times D_5$ (as 20T6)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 + x^17 - x^15 - 4*x^14 + 11*x^13 - 6*x^12 - 8*x^11 + 22*x^10 - 17*x^9 + 9*x^8 - 31*x^7 + 56*x^6 - 64*x^5 + 27*x^4 + 10*x^3 + 4*x^2 + x + 1)
 
Copy content gp:K = bnfinit(y^20 - 2*y^19 + y^18 + y^17 - y^15 - 4*y^14 + 11*y^13 - 6*y^12 - 8*y^11 + 22*y^10 - 17*y^9 + 9*y^8 - 31*y^7 + 56*y^6 - 64*y^5 + 27*y^4 + 10*y^3 + 4*y^2 + y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + x^18 + x^17 - x^15 - 4*x^14 + 11*x^13 - 6*x^12 - 8*x^11 + 22*x^10 - 17*x^9 + 9*x^8 - 31*x^7 + 56*x^6 - 64*x^5 + 27*x^4 + 10*x^3 + 4*x^2 + x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 2*x^19 + x^18 + x^17 - x^15 - 4*x^14 + 11*x^13 - 6*x^12 - 8*x^11 + 22*x^10 - 17*x^9 + 9*x^8 - 31*x^7 + 56*x^6 - 64*x^5 + 27*x^4 + 10*x^3 + 4*x^2 + x + 1)
 

\( x^{20} - 2 x^{19} + x^{18} + x^{17} - x^{15} - 4 x^{14} + 11 x^{13} - 6 x^{12} - 8 x^{11} + 22 x^{10} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 10)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(386587549251591827392578125\) \(\medspace = 5^{15}\cdot 103^{8}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(21.35\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $5^{3/4}103^{1/2}\approx 33.93486420206204$
Ramified primes:   \(5\), \(103\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\zeta_{5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{54391}a^{17}+\frac{27078}{54391}a^{16}-\frac{8962}{54391}a^{15}-\frac{8922}{54391}a^{14}+\frac{17886}{54391}a^{13}-\frac{22579}{54391}a^{12}+\frac{22588}{54391}a^{11}+\frac{24920}{54391}a^{10}-\frac{22284}{54391}a^{9}-\frac{3100}{54391}a^{8}+\frac{2659}{54391}a^{7}+\frac{3989}{54391}a^{6}-\frac{13595}{54391}a^{5}-\frac{11938}{54391}a^{4}-\frac{7456}{54391}a^{3}-\frac{9815}{54391}a^{2}+\frac{5260}{54391}a-\frac{22604}{54391}$, $\frac{1}{54391}a^{18}+\frac{18025}{54391}a^{16}+\frac{25863}{54391}a^{15}+\frac{2980}{54391}a^{14}+\frac{12168}{54391}a^{13}+\frac{7519}{54391}a^{12}+\frac{13851}{54391}a^{11}+\frac{23093}{54391}a^{10}-\frac{10702}{54391}a^{9}+\frac{19146}{54391}a^{8}+\frac{17271}{54391}a^{7}-\frac{7211}{54391}a^{6}-\frac{4816}{54391}a^{5}+\frac{3995}{54391}a^{4}-\frac{15639}{54391}a^{3}+\frac{21404}{54391}a^{2}-\frac{2855}{54391}a+\frac{9189}{54391}$, $\frac{1}{54391}a^{19}-\frac{4644}{54391}a^{16}+\frac{1760}{54391}a^{15}-\frac{2969}{54391}a^{14}-\frac{12174}{54391}a^{13}-\frac{7527}{54391}a^{12}-\frac{8972}{54391}a^{11}+\frac{21567}{54391}a^{10}+\frac{10711}{54391}a^{9}-\frac{19177}{54391}a^{8}-\frac{17215}{54391}a^{7}-\frac{1639}{54391}a^{6}+\frac{22415}{54391}a^{5}-\frac{3985}{54391}a^{4}+\frac{15643}{54391}a^{3}-\frac{21403}{54391}a^{2}+\frac{1202}{54391}a-\frac{5881}{54391}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -\frac{7632}{54391} a^{19} + \frac{23286}{54391} a^{18} - \frac{22896}{54391} a^{17} + \frac{7632}{54391} a^{15} + \frac{7632}{54391} a^{14} + \frac{25452}{54391} a^{13} - \frac{114480}{54391} a^{12} + \frac{129744}{54391} a^{11} + \frac{15264}{54391} a^{10} - \frac{228960}{54391} a^{9} + \frac{307103}{54391} a^{8} - \frac{198432}{54391} a^{7} + \frac{305280}{54391} a^{6} - \frac{663984}{54391} a^{5} + \frac{915840}{54391} a^{4} - \frac{694605}{54391} a^{3} + \frac{129744}{54391} a^{2} + \frac{45792}{54391} a + \frac{22896}{54391} \)  (order $10$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{19848}{54391}a^{19}-\frac{41102}{54391}a^{18}+\frac{20551}{54391}a^{17}+\frac{20551}{54391}a^{16}-\frac{22648}{54391}a^{14}-\frac{82204}{54391}a^{13}+\frac{226061}{54391}a^{12}-\frac{123306}{54391}a^{11}-\frac{164408}{54391}a^{10}+\frac{439960}{54391}a^{9}-\frac{349367}{54391}a^{8}+\frac{184959}{54391}a^{7}-\frac{637081}{54391}a^{6}+\frac{1150856}{54391}a^{5}-\frac{1291248}{54391}a^{4}+\frac{554877}{54391}a^{3}+\frac{205510}{54391}a^{2}+\frac{82204}{54391}a-\frac{33840}{54391}$, $\frac{8400}{54391}a^{19}-\frac{8400}{54391}a^{18}-\frac{5600}{54391}a^{17}+\frac{8599}{54391}a^{16}+\frac{16800}{54391}a^{15}-\frac{8400}{54391}a^{14}-\frac{44800}{54391}a^{13}+\frac{56000}{54391}a^{12}+\frac{35741}{54391}a^{11}-\frac{75600}{54391}a^{10}+\frac{70000}{54391}a^{9}+\frac{36400}{54391}a^{8}+\frac{16800}{54391}a^{7}-\frac{281505}{54391}a^{6}+\frac{282800}{54391}a^{5}-\frac{179200}{54391}a^{4}-\frac{67200}{54391}a^{3}-\frac{25200}{54391}a^{2}+\frac{380302}{54391}a-\frac{5600}{54391}$, $\frac{114}{54391}a^{19}-\frac{57}{54391}a^{18}-\frac{57}{54391}a^{17}-\frac{1243}{54391}a^{15}+\frac{228}{54391}a^{14}-\frac{627}{54391}a^{13}+\frac{342}{54391}a^{12}+\frac{456}{54391}a^{11}-\frac{6148}{54391}a^{10}+\frac{969}{54391}a^{9}-\frac{513}{54391}a^{8}+\frac{1767}{54391}a^{7}-\frac{3192}{54391}a^{6}-\frac{39460}{54391}a^{5}-\frac{1539}{54391}a^{4}-\frac{570}{54391}a^{3}-\frac{228}{54391}a^{2}-\frac{57}{54391}a+\frac{39830}{54391}$, $\frac{23997}{54391}a^{19}-\frac{36156}{54391}a^{18}-\frac{959}{54391}a^{17}+\frac{37249}{54391}a^{16}+\frac{15528}{54391}a^{15}-\frac{28772}{54391}a^{14}-\frac{111117}{54391}a^{13}+\frac{219553}{54391}a^{12}-\frac{208}{54391}a^{11}-\frac{273919}{54391}a^{10}+\frac{413760}{54391}a^{9}-\frac{124124}{54391}a^{8}+\frac{10481}{54391}a^{7}-\frac{652310}{54391}a^{6}+\frac{1004664}{54391}a^{5}-\frac{833215}{54391}a^{4}-\frac{164853}{54391}a^{3}+\frac{491007}{54391}a^{2}+\frac{294404}{54391}a+\frac{30650}{54391}$, $\frac{30926}{54391}a^{19}-\frac{555}{499}a^{18}+\frac{23331}{54391}a^{17}+\frac{40033}{54391}a^{16}+\frac{272}{54391}a^{15}-\frac{35437}{54391}a^{14}-\frac{126647}{54391}a^{13}+\frac{336038}{54391}a^{12}-\frac{145024}{54391}a^{11}-\frac{297216}{54391}a^{10}+\frac{676870}{54391}a^{9}-\frac{444952}{54391}a^{8}+\frac{170154}{54391}a^{7}-\frac{901966}{54391}a^{6}+\frac{1673816}{54391}a^{5}-\frac{1738095}{54391}a^{4}+\frac{556507}{54391}a^{3}+\frac{562713}{54391}a^{2}+\frac{115154}{54391}a-\frac{58016}{54391}$, $\frac{15776}{54391}a^{19}-\frac{32824}{54391}a^{18}+\frac{14959}{54391}a^{17}+\frac{23656}{54391}a^{16}-\frac{9168}{54391}a^{15}-\frac{17479}{54391}a^{14}-\frac{58768}{54391}a^{13}+\frac{184325}{54391}a^{12}-\frac{99527}{54391}a^{11}-\frac{167176}{54391}a^{10}+\frac{406089}{54391}a^{9}-\frac{277632}{54391}a^{8}+\frac{75063}{54391}a^{7}-\frac{413120}{54391}a^{6}+\frac{859496}{54391}a^{5}-\frac{926079}{54391}a^{4}+\frac{243480}{54391}a^{3}+\frac{423103}{54391}a^{2}-\frac{151099}{54391}a+\frac{60751}{54391}$, $\frac{549}{54391}a^{19}+\frac{979}{54391}a^{18}-\frac{8847}{54391}a^{17}+\frac{9510}{54391}a^{16}-\frac{6352}{54391}a^{14}-\frac{6613}{54391}a^{13}-\frac{2021}{54391}a^{12}+\frac{37221}{54391}a^{11}-\frac{56081}{54391}a^{10}+\frac{11}{109}a^{9}+\frac{69747}{54391}a^{8}-\frac{130336}{54391}a^{7}+\frac{45173}{54391}a^{6}-\frac{61696}{54391}a^{5}+\frac{188556}{54391}a^{4}-\frac{317487}{54391}a^{3}+\frac{309414}{54391}a^{2}+\frac{9578}{54391}a-\frac{16203}{54391}$, $\frac{12777}{54391}a^{19}-\frac{23189}{54391}a^{18}+\frac{6482}{54391}a^{17}+\frac{17142}{54391}a^{16}-\frac{11416}{54391}a^{14}-\frac{51133}{54391}a^{13}+\frac{128149}{54391}a^{12}-\frac{50547}{54391}a^{11}-\frac{126041}{54391}a^{10}+\frac{278799}{54391}a^{9}-\frac{163029}{54391}a^{8}+\frac{32945}{54391}a^{7}-\frac{342787}{54391}a^{6}+\frac{630272}{54391}a^{5}-\frac{654966}{54391}a^{4}+\frac{144249}{54391}a^{3}+\frac{335322}{54391}a^{2}-\frac{31788}{54391}a+\frac{2877}{54391}$, $\frac{10261}{54391}a^{19}-\frac{18206}{54391}a^{18}+\frac{5905}{54391}a^{17}+\frac{12626}{54391}a^{16}+\frac{4490}{54391}a^{15}-\frac{11533}{54391}a^{14}-\frac{42235}{54391}a^{13}+\frac{104034}{54391}a^{12}-\frac{31342}{54391}a^{11}-\frac{90557}{54391}a^{10}+\frac{195783}{54391}a^{9}-\frac{107622}{54391}a^{8}+\frac{54185}{54391}a^{7}-\frac{295581}{54391}a^{6}+\frac{529131}{54391}a^{5}-\frac{547320}{54391}a^{4}+\frac{183934}{54391}a^{3}+\frac{68197}{54391}a^{2}+\frac{133511}{54391}a-\frac{14646}{54391}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 661565.5195213907 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 661565.5195213907 \cdot 1}{10\cdot\sqrt{386587549251591827392578125}}\cr\approx \mathstrut & 0.322661846006739 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 + x^17 - x^15 - 4*x^14 + 11*x^13 - 6*x^12 - 8*x^11 + 22*x^10 - 17*x^9 + 9*x^8 - 31*x^7 + 56*x^6 - 64*x^5 + 27*x^4 + 10*x^3 + 4*x^2 + x + 1) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 2*x^19 + x^18 + x^17 - x^15 - 4*x^14 + 11*x^13 - 6*x^12 - 8*x^11 + 22*x^10 - 17*x^9 + 9*x^8 - 31*x^7 + 56*x^6 - 64*x^5 + 27*x^4 + 10*x^3 + 4*x^2 + x + 1, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + x^18 + x^17 - x^15 - 4*x^14 + 11*x^13 - 6*x^12 - 8*x^11 + 22*x^10 - 17*x^9 + 9*x^8 - 31*x^7 + 56*x^6 - 64*x^5 + 27*x^4 + 10*x^3 + 4*x^2 + x + 1); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 2*x^19 + x^18 + x^17 - x^15 - 4*x^14 + 11*x^13 - 6*x^12 - 8*x^11 + 22*x^10 - 17*x^9 + 9*x^8 - 31*x^7 + 56*x^6 - 64*x^5 + 27*x^4 + 10*x^3 + 4*x^2 + x + 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_4\times D_5$ (as 20T6):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.10609.1, 10.2.351721503125.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 40
Degree 20 sibling: deg 20
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $20$ ${\href{/padicField/3.4.0.1}{4} }^{5}$ R $20$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ ${\href{/padicField/47.4.0.1}{4} }^{5}$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(103\) Copy content Toggle raw display 103.4.1.0a1.1$x^{4} + 2 x^{2} + 88 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
103.4.2.4a1.2$x^{8} + 4 x^{6} + 176 x^{5} + 14 x^{4} + 352 x^{3} + 7764 x^{2} + 880 x + 128$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
103.4.2.4a1.2$x^{8} + 4 x^{6} + 176 x^{5} + 14 x^{4} + 352 x^{3} + 7764 x^{2} + 880 x + 128$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*40 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*40 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.515.2t1.a.a$1$ $ 5 \cdot 103 $ \(\Q(\sqrt{-515}) \) $C_2$ (as 2T1) $1$ $-1$
1.103.2t1.a.a$1$ $ 103 $ \(\Q(\sqrt{-103}) \) $C_2$ (as 2T1) $1$ $-1$
1.515.4t1.a.a$1$ $ 5 \cdot 103 $ \(\Q(\sqrt{1030 -206 \sqrt{5}})\) $C_4$ (as 4T1) $0$ $1$
*40 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
*40 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.515.4t1.a.b$1$ $ 5 \cdot 103 $ \(\Q(\sqrt{1030 -206 \sqrt{5}})\) $C_4$ (as 4T1) $0$ $1$
*40 2.2575.10t3.b.b$2$ $ 5^{2} \cdot 103 $ 10.0.36227314821875.1 $D_{10}$ (as 10T3) $1$ $0$
*40 2.103.5t2.a.a$2$ $ 103 $ 5.1.10609.1 $D_{5}$ (as 5T2) $1$ $0$
*40 2.2575.10t3.b.a$2$ $ 5^{2} \cdot 103 $ 10.0.36227314821875.1 $D_{10}$ (as 10T3) $1$ $0$
*40 2.103.5t2.a.b$2$ $ 103 $ 5.1.10609.1 $D_{5}$ (as 5T2) $1$ $0$
*40 2.2575.20t6.a.b$2$ $ 5^{2} \cdot 103 $ 20.0.386587549251591827392578125.1 $C_4\times D_5$ (as 20T6) $0$ $0$
*40 2.2575.20t6.a.d$2$ $ 5^{2} \cdot 103 $ 20.0.386587549251591827392578125.1 $C_4\times D_5$ (as 20T6) $0$ $0$
*40 2.2575.20t6.a.a$2$ $ 5^{2} \cdot 103 $ 20.0.386587549251591827392578125.1 $C_4\times D_5$ (as 20T6) $0$ $0$
*40 2.2575.20t6.a.c$2$ $ 5^{2} \cdot 103 $ 20.0.386587549251591827392578125.1 $C_4\times D_5$ (as 20T6) $0$ $0$

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 *.

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)