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Properties

Label 15.3.21643467887730481.1
Degree $15$
Signature $(3, 6)$
Discriminant $2.164\times 10^{16}$
Root discriminant \(12.28\)
Ramified primes $13,47,109$
Class number $1$
Class group trivial
Galois group $C_3\wr D_5$ (as 15T46)

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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 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*x - 1)
 
Copy content gp:K = bnfinit(y^15 - y^14 - y^13 + y^12 - 7*y^11 - 2*y^10 + 18*y^9 + 12*y^8 - 17*y^7 + 7*y^6 - 11*y^5 - 21*y^4 + 22*y^3 - 11*y^2 - 2*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*x - 1)
 

\( x^{15} - x^{14} - x^{13} + x^{12} - 7 x^{11} - 2 x^{10} + 18 x^{9} + 12 x^{8} - 17 x^{7} + 7 x^{6} + \cdots - 1 \) 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:  $(3, 6)$
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:   \(21643467887730481\) \(\medspace = 13^{2}\cdot 47^{6}\cdot 109^{2}\) 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:  \(12.28\)
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:  $13^{2/3}47^{1/2}109^{2/3}\approx 864.8925624711881$
Ramified primes:   \(13\), \(47\), \(109\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_3$
Copy content comment:Autmorphisms
 
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4823316529}a^{14}-\frac{240451914}{4823316529}a^{13}-\frac{1467210693}{4823316529}a^{12}+\frac{2044776038}{4823316529}a^{11}+\frac{2380646120}{4823316529}a^{10}-\frac{2290615831}{4823316529}a^{9}-\frac{1561717558}{4823316529}a^{8}+\frac{2080171999}{4823316529}a^{7}-\frac{2386498101}{4823316529}a^{6}+\frac{1715950732}{4823316529}a^{5}+\frac{1834453739}{4823316529}a^{4}+\frac{64937028}{4823316529}a^{3}+\frac{593205418}{4823316529}a^{2}+\frac{970975456}{4823316529}a+\frac{820723294}{4823316529}$ 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:  $8$
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:   $\frac{438282190}{4823316529}a^{14}-\frac{666156758}{4823316529}a^{13}-\frac{180199038}{4823316529}a^{12}+\frac{937506764}{4823316529}a^{11}-\frac{3215866687}{4823316529}a^{10}+\frac{764978875}{4823316529}a^{9}+\frac{8142107838}{4823316529}a^{8}-\frac{819825977}{4823316529}a^{7}-\frac{12445039424}{4823316529}a^{6}+\frac{7101345595}{4823316529}a^{5}-\frac{2075689344}{4823316529}a^{4}-\frac{3242761291}{4823316529}a^{3}+\frac{18275305195}{4823316529}a^{2}-\frac{11471590121}{4823316529}a-\frac{1755151205}{4823316529}$, $\frac{279318677}{4823316529}a^{14}-\frac{570733139}{4823316529}a^{13}-\frac{97673519}{4823316529}a^{12}+\frac{286289673}{4823316529}a^{11}-\frac{2256835527}{4823316529}a^{10}+\frac{1369023982}{4823316529}a^{9}+\frac{6081306034}{4823316529}a^{8}+\frac{1059190063}{4823316529}a^{7}-\frac{6065350387}{4823316529}a^{6}+\frac{5660375798}{4823316529}a^{5}-\frac{6997088601}{4823316529}a^{4}-\frac{5291197834}{4823316529}a^{3}+\frac{4910463282}{4823316529}a^{2}-\frac{11751794763}{4823316529}a+\frac{4235471514}{4823316529}$, $\frac{38028855}{4823316529}a^{14}+\frac{30131136}{4823316529}a^{13}-\frac{41830710}{4823316529}a^{12}+\frac{95826109}{4823316529}a^{11}+\frac{13051036}{4823316529}a^{10}-\frac{448791598}{4823316529}a^{9}+\frac{291200427}{4823316529}a^{8}+\frac{903950089}{4823316529}a^{7}-\frac{1148631260}{4823316529}a^{6}-\frac{1349674114}{4823316529}a^{5}+\frac{1574584533}{4823316529}a^{4}+\frac{638893288}{4823316529}a^{3}-\frac{3508838763}{4823316529}a^{2}+\frac{1850995278}{4823316529}a+\frac{1978357973}{4823316529}$, $\frac{566139539}{4823316529}a^{14}-\frac{325368897}{4823316529}a^{13}-\frac{420696872}{4823316529}a^{12}+\frac{578463500}{4823316529}a^{11}-\frac{3451205678}{4823316529}a^{10}-\frac{2747557933}{4823316529}a^{9}+\frac{7365442564}{4823316529}a^{8}+\frac{6303751499}{4823316529}a^{7}-\frac{9186881148}{4823316529}a^{6}+\frac{2757048777}{4823316529}a^{5}-\frac{662352361}{4823316529}a^{4}-\frac{5601051546}{4823316529}a^{3}+\frac{12324635935}{4823316529}a^{2}+\frac{209859584}{4823316529}a-\frac{2375337177}{4823316529}$, $\frac{469621379}{4823316529}a^{14}+\frac{58489305}{4823316529}a^{13}-\frac{825121412}{4823316529}a^{12}+\frac{90755217}{4823316529}a^{11}-\frac{2804712262}{4823316529}a^{10}-\frac{4403499472}{4823316529}a^{9}+\frac{6154418487}{4823316529}a^{8}+\frac{12709698685}{4823316529}a^{7}-\frac{2583762033}{4823316529}a^{6}-\frac{4750900235}{4823316529}a^{5}-\frac{1059112278}{4823316529}a^{4}-\frac{12098985853}{4823316529}a^{3}+\frac{4091725504}{4823316529}a^{2}+\frac{10667639529}{4823316529}a-\frac{2580378452}{4823316529}$, $\frac{1205387739}{4823316529}a^{14}-\frac{816773200}{4823316529}a^{13}-\frac{1373326882}{4823316529}a^{12}+\frac{706262859}{4823316529}a^{11}-\frac{7858948337}{4823316529}a^{10}-\frac{4862149836}{4823316529}a^{9}+\frac{19365056971}{4823316529}a^{8}+\frac{20389332875}{4823316529}a^{7}-\frac{14912893927}{4823316529}a^{6}+\frac{1147428040}{4823316529}a^{5}-\frac{12087988177}{4823316529}a^{4}-\frac{21912285059}{4823316529}a^{3}+\frac{18575332517}{4823316529}a^{2}-\frac{6222009777}{4823316529}a+\frac{700795973}{4823316529}$, $\frac{667132927}{4823316529}a^{14}-\frac{753110120}{4823316529}a^{13}-\frac{1017644411}{4823316529}a^{12}+\frac{516490886}{4823316529}a^{11}-\frac{4405300094}{4823316529}a^{10}-\frac{813451459}{4823316529}a^{9}+\frac{14786449294}{4823316529}a^{8}+\frac{11452381693}{4823316529}a^{7}-\frac{13822873461}{4823316529}a^{6}-\frac{3214737208}{4823316529}a^{5}-\frac{9422680395}{4823316529}a^{4}-\frac{15211313221}{4823316529}a^{3}+\frac{15929636186}{4823316529}a^{2}-\frac{4384209022}{4823316529}a-\frac{4901669196}{4823316529}$, $\frac{648967434}{4823316529}a^{14}-\frac{1114737512}{4823316529}a^{13}-\frac{747311498}{4823316529}a^{12}+\frac{1124779681}{4823316529}a^{11}-\frac{4823766233}{4823316529}a^{10}+\frac{1779201762}{4823316529}a^{9}+\frac{15799224682}{4823316529}a^{8}+\frac{4012111589}{4823316529}a^{7}-\frac{20004131378}{4823316529}a^{6}+\frac{3648655910}{4823316529}a^{5}-\frac{7466582441}{4823316529}a^{4}-\frac{13931973140}{4823316529}a^{3}+\frac{24631797650}{4823316529}a^{2}-\frac{10717944600}{4823316529}a-\frac{879206783}{4823316529}$ 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:  \( 116.633322592 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 116.633322592 \cdot 1}{2\cdot\sqrt{21643467887730481}}\cr\approx \mathstrut & 0.195118489473 \end{aligned}\]

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 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 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^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 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^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 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^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 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

$C_3\wr D_5$ (as 15T46):

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 2430
The 72 conjugacy class representatives for $C_3\wr D_5$
Character table for $C_3\wr D_5$

Intermediate fields

5.1.2209.1

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

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 siblings: data not computed
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 $15$ ${\href{/padicField/3.5.0.1}{5} }^{3}$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ $15$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ R $15$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ $15$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ R $15$ ${\href{/padicField/59.5.0.1}{5} }^{3}$

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
\(13\) Copy content Toggle raw display 13.1.3.2a1.1$x^{3} + 13$$3$$1$$2$$C_3$$$[\ ]_{3}$$
13.6.1.0a1.1$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
13.6.1.0a1.1$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(47\) Copy content Toggle raw display 47.3.1.0a1.1$x^{3} + 3 x + 42$$1$$3$$0$$C_3$$$[\ ]^{3}$$
47.3.2.3a1.2$x^{6} + 6 x^{4} + 84 x^{3} + 9 x^{2} + 252 x + 1811$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
47.3.2.3a1.2$x^{6} + 6 x^{4} + 84 x^{3} + 9 x^{2} + 252 x + 1811$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(109\) Copy content Toggle raw display 109.1.3.2a1.2$x^{3} + 654$$3$$1$$2$$C_3$$$[\ ]_{3}$$
109.6.1.0a1.1$x^{6} + 107 x^{3} + 102 x^{2} + 66 x + 6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
109.6.1.0a1.1$x^{6} + 107 x^{3} + 102 x^{2} + 66 x + 6$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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