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Properties

Label 15.5.244...591.1
Degree $15$
Signature $[5, 5]$
Discriminant $-2.442\times 10^{19}$
Root discriminant \(19.61\)
Ramified prime $31$
Class number $1$
Class group trivial
Galois group $S_3 \times C_5$ (as 15T4)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*x - 1)
 
Copy content gp:K = bnfinit(y^15 - 4*y^14 + 7*y^13 - 17*y^12 + 51*y^11 - 87*y^10 - 7*y^9 + 172*y^8 - 100*y^7 - 69*y^6 - 17*y^5 + 62*y^4 + 50*y^3 - 22*y^2 - 18*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*x - 1)
 

\( x^{15} - 4 x^{14} + 7 x^{13} - 17 x^{12} + 51 x^{11} - 87 x^{10} - 7 x^{9} + 172 x^{8} - 100 x^{7} + \cdots - 1 \) 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:  $15$
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:  $[5, 5]$
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:   \(-24417546297445042591\) \(\medspace = -\,31^{13}\) 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:  \(19.61\)
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:  $31^{9/10}\approx 21.990014941549536$
Ramified primes:   \(31\) 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:  \(\Q(\sqrt{-31}) \)
$\Aut(K/\Q)$:   $C_5$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{67}a^{13}-\frac{30}{67}a^{12}+\frac{15}{67}a^{11}-\frac{27}{67}a^{10}+\frac{27}{67}a^{9}+\frac{22}{67}a^{8}+\frac{17}{67}a^{7}+\frac{32}{67}a^{6}+\frac{14}{67}a^{5}-\frac{12}{67}a^{4}+\frac{6}{67}a^{3}-\frac{9}{67}a^{2}+\frac{7}{67}a-\frac{23}{67}$, $\frac{1}{2328257544133}a^{14}+\frac{16975831322}{2328257544133}a^{13}+\frac{935820820203}{2328257544133}a^{12}-\frac{434972844569}{2328257544133}a^{11}-\frac{814302805997}{2328257544133}a^{10}-\frac{211932806836}{2328257544133}a^{9}+\frac{12371814752}{2328257544133}a^{8}-\frac{357482693880}{2328257544133}a^{7}-\frac{6663952029}{2328257544133}a^{6}-\frac{683853589830}{2328257544133}a^{5}-\frac{1156621051070}{2328257544133}a^{4}+\frac{132213504029}{2328257544133}a^{3}-\frac{1154630429440}{2328257544133}a^{2}-\frac{892730603990}{2328257544133}a+\frac{1076347464613}{2328257544133}$ 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 $1$
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 $1$
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:   \( -1 \)  (order $2$) 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:   $\frac{243059409}{2328257544133}a^{14}-\frac{58795611972}{2328257544133}a^{13}+\frac{283835217917}{2328257544133}a^{12}-\frac{671174577289}{2328257544133}a^{11}+\frac{1545391784548}{2328257544133}a^{10}-\frac{4167500959675}{2328257544133}a^{9}+\frac{8618200703888}{2328257544133}a^{8}-\frac{6698796274221}{2328257544133}a^{7}-\frac{6040944747651}{2328257544133}a^{6}+\frac{14805483642160}{2328257544133}a^{5}-\frac{5586924398371}{2328257544133}a^{4}-\frac{724165140729}{2328257544133}a^{3}-\frac{5814004335306}{2328257544133}a^{2}+\frac{4159657329275}{2328257544133}a+\frac{3685669397498}{2328257544133}$, $\frac{267585097919}{2328257544133}a^{14}-\frac{1253520182462}{2328257544133}a^{13}+\frac{2641390360838}{2328257544133}a^{12}-\frac{6138719447285}{2328257544133}a^{11}+\frac{17569473744530}{2328257544133}a^{10}-\frac{34144131166079}{2328257544133}a^{9}+\frac{18485848976355}{2328257544133}a^{8}+\frac{36794259919607}{2328257544133}a^{7}-\frac{47010188618512}{2328257544133}a^{6}+\frac{9288455272432}{2328257544133}a^{5}-\frac{14348444844804}{2328257544133}a^{4}+\frac{23900361256751}{2328257544133}a^{3}+\frac{6093122570911}{2328257544133}a^{2}-\frac{4109472049064}{2328257544133}a-\frac{2536176204826}{2328257544133}$, $\frac{107616096499}{2328257544133}a^{14}-\frac{279921774352}{2328257544133}a^{13}+\frac{273579756429}{2328257544133}a^{12}-\frac{1174327988520}{2328257544133}a^{11}+\frac{3541136186138}{2328257544133}a^{10}-\frac{3439217257266}{2328257544133}a^{9}-\frac{8823909868513}{2328257544133}a^{8}+\frac{10130838537873}{2328257544133}a^{7}+\frac{10141573761668}{2328257544133}a^{6}-\frac{5961882724531}{2328257544133}a^{5}-\frac{18352065081316}{2328257544133}a^{4}-\frac{664268199043}{2328257544133}a^{3}+\frac{11897335245225}{2328257544133}a^{2}+\frac{3226043066405}{2328257544133}a-\frac{827874003385}{2328257544133}$, $\frac{15534851802}{2328257544133}a^{14}-\frac{310420367835}{2328257544133}a^{13}+\frac{846059759719}{2328257544133}a^{12}-\frac{1224851620432}{2328257544133}a^{11}+\frac{4012431435855}{2328257544133}a^{10}-\frac{10833583530070}{2328257544133}a^{9}+\frac{11924086802913}{2328257544133}a^{8}+\frac{16551040214255}{2328257544133}a^{7}-\frac{28146605200127}{2328257544133}a^{6}-\frac{9744470481690}{2328257544133}a^{5}+\frac{11980428858427}{2328257544133}a^{4}+\frac{27406648660638}{2328257544133}a^{3}-\frac{1611692855579}{2328257544133}a^{2}-\frac{13921249553043}{2328257544133}a-\frac{3308306735870}{2328257544133}$, $\frac{221069717989}{2328257544133}a^{14}-\frac{812803982821}{2328257544133}a^{13}+\frac{1441695630178}{2328257544133}a^{12}-\frac{3830724914016}{2328257544133}a^{11}+\frac{10919146009939}{2328257544133}a^{10}-\frac{18122667063096}{2328257544133}a^{9}-\frac{453458022259}{2328257544133}a^{8}+\frac{27069382635229}{2328257544133}a^{7}-\frac{17519724693137}{2328257544133}a^{6}+\frac{69067802221}{2328257544133}a^{5}-\frac{13704436043447}{2328257544133}a^{4}+\frac{4042596563740}{2328257544133}a^{3}+\frac{4776557664914}{2328257544133}a^{2}+\frac{1488718191345}{2328257544133}a+\frac{3046632114016}{2328257544133}$, $\frac{106584952006}{2328257544133}a^{14}-\frac{459656461286}{2328257544133}a^{13}+\frac{1012465523189}{2328257544133}a^{12}-\frac{2613992273846}{2328257544133}a^{11}+\frac{7131863600067}{2328257544133}a^{10}-\frac{13756623825798}{2328257544133}a^{9}+\frac{10104727394380}{2328257544133}a^{8}+\frac{3923788363950}{2328257544133}a^{7}-\frac{10441282622150}{2328257544133}a^{6}+\frac{11823397836943}{2328257544133}a^{5}-\frac{15073170859941}{2328257544133}a^{4}+\frac{9596079935112}{2328257544133}a^{3}-\frac{2226004000257}{2328257544133}a^{2}-\frac{1474073828694}{2328257544133}a+\frac{942994534295}{2328257544133}$, $\frac{1151156254397}{2328257544133}a^{14}-\frac{4823448537837}{2328257544133}a^{13}+\frac{9134091304009}{2328257544133}a^{12}-\frac{21865623729734}{2328257544133}a^{11}+\frac{63766691902375}{2328257544133}a^{10}-\frac{114734559068094}{2328257544133}a^{9}+\frac{20949315679253}{2328257544133}a^{8}+\frac{182877151681740}{2328257544133}a^{7}-\frac{154002735802108}{2328257544133}a^{6}-\frac{28038620815650}{2328257544133}a^{5}-\frac{22617506148656}{2328257544133}a^{4}+\frac{69907726933390}{2328257544133}a^{3}+\frac{37427374092455}{2328257544133}a^{2}-\frac{29479559531629}{2328257544133}a-\frac{9738854803650}{2328257544133}$, $\frac{1049496746131}{2328257544133}a^{14}-\frac{4249575155725}{2328257544133}a^{13}+\frac{8000993702770}{2328257544133}a^{12}-\frac{19776577180236}{2328257544133}a^{11}+\frac{56975492704171}{2328257544133}a^{10}-\frac{100984549198382}{2328257544133}a^{9}+\frac{17616560734730}{2328257544133}a^{8}+\frac{148832776558315}{2328257544133}a^{7}-\frac{123755117658611}{2328257544133}a^{6}-\frac{8095199139875}{2328257544133}a^{5}-\frac{37055740357823}{2328257544133}a^{4}+\frac{49720924501894}{2328257544133}a^{3}+\frac{27438573459884}{2328257544133}a^{2}-\frac{17514774224470}{2328257544133}a-\frac{659456866622}{2328257544133}$, $\frac{292952766669}{2328257544133}a^{14}-\frac{834149796827}{2328257544133}a^{13}+\frac{1144564268545}{2328257544133}a^{12}-\frac{3951254014228}{2328257544133}a^{11}+\frac{11055772575791}{2328257544133}a^{10}-\frac{14236867825707}{2328257544133}a^{9}-\frac{14330025219917}{2328257544133}a^{8}+\frac{25327746559616}{2328257544133}a^{7}+\frac{6359205033623}{2328257544133}a^{6}-\frac{4692446767528}{2328257544133}a^{5}-\frac{26613128068049}{2328257544133}a^{4}-\frac{5299438870211}{2328257544133}a^{3}+\frac{10542446704233}{2328257544133}a^{2}+\frac{6876427063067}{2328257544133}a+\frac{1823742076844}{2328257544133}$ Copy content Toggle raw display
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:  \( 6179.9389969906615 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 6179.9389969906615 \cdot 1}{2\cdot\sqrt{24417546297445042591}}\cr\approx \mathstrut & 0.195953264325999 \end{aligned}\]

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^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*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^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*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^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*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^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*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

$C_5\times S_3$ (as 15T4):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
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 30
The 15 conjugacy class representatives for $S_3 \times C_5$
Character table for $S_3 \times C_5$

Intermediate fields

3.1.31.1, 5.5.923521.1

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 30
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 $15$ ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ ${\href{/padicField/5.3.0.1}{3} }^{5}$ $15$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ $15$ ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ R ${\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{5}$ $15$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ ${\href{/padicField/47.5.0.1}{5} }^{3}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ $15$

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
\(31\) Copy content Toggle raw display 31.1.5.4a1.1$x^{5} + 31$$5$$1$$4$$C_5$$$[\ ]_{5}$$
31.1.10.9a1.1$x^{10} + 31$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*30 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.31.2t1.a.a$1$ $ 31 $ \(\Q(\sqrt{-31}) \) $C_2$ (as 2T1) $1$ $-1$
1.31.10t1.a.a$1$ $ 31 $ 10.0.26439622160671.1 $C_{10}$ (as 10T1) $0$ $-1$
*30 1.31.5t1.a.a$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
*30 1.31.5t1.a.b$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
1.31.10t1.a.b$1$ $ 31 $ 10.0.26439622160671.1 $C_{10}$ (as 10T1) $0$ $-1$
1.31.10t1.a.c$1$ $ 31 $ 10.0.26439622160671.1 $C_{10}$ (as 10T1) $0$ $-1$
*30 1.31.5t1.a.c$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
*30 1.31.5t1.a.d$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
1.31.10t1.a.d$1$ $ 31 $ 10.0.26439622160671.1 $C_{10}$ (as 10T1) $0$ $-1$
*30 2.31.3t2.b.a$2$ $ 31 $ 3.1.31.1 $S_3$ (as 3T2) $1$ $0$
*30 2.961.15t4.a.a$2$ $ 31^{2}$ 15.5.24417546297445042591.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
*30 2.961.15t4.a.b$2$ $ 31^{2}$ 15.5.24417546297445042591.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
*30 2.961.15t4.a.c$2$ $ 31^{2}$ 15.5.24417546297445042591.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
*30 2.961.15t4.a.d$2$ $ 31^{2}$ 15.5.24417546297445042591.1 $S_3 \times C_5$ (as 15T4) $0$ $0$

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)