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Properties

Label 15.3.12205980432301041.1
Degree $15$
Signature $[3, 6]$
Discriminant $1.221\times 10^{16}$
Root discriminant \(11.82\)
Ramified primes $3,1871$
Class number $1$
Class group trivial
Galois group $D_5\wr C_3$ (as 15T50)

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

\( x^{15} - 3x^{14} + 7x^{12} - 6x^{11} + 4x^{9} - 6x^{8} + 6x^{7} - 7x^{6} + 6x^{4} - 5x^{3} + 3x^{2} + 1 \) 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:   \(12205980432301041\) \(\medspace = 3^{20}\cdot 1871^{2}\) 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:  \(11.82\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{4/3}1871^{1/2}\approx 187.15376431258608$
Ramified primes:   \(3\), \(1871\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{323}a^{14}+\frac{44}{323}a^{13}+\frac{130}{323}a^{12}-\frac{20}{323}a^{11}+\frac{23}{323}a^{10}+\frac{112}{323}a^{9}+\frac{100}{323}a^{8}-\frac{151}{323}a^{7}+\frac{15}{323}a^{6}+\frac{52}{323}a^{5}-\frac{140}{323}a^{4}-\frac{6}{17}a^{3}+\frac{128}{323}a^{2}-\frac{118}{323}a-\frac{55}{323}$ 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:   $a$, $\frac{1}{17}a^{14}+\frac{7}{17}a^{13}-\frac{11}{17}a^{12}-\frac{14}{17}a^{11}+\frac{45}{17}a^{10}-\frac{10}{17}a^{9}-\frac{49}{17}a^{8}+\frac{32}{17}a^{7}+\frac{2}{17}a^{6}+\frac{16}{17}a^{5}+\frac{4}{17}a^{4}-\frac{39}{17}a^{3}+\frac{25}{17}a^{2}-\frac{1}{17}a-\frac{30}{17}$, $\frac{398}{323}a^{14}-\frac{899}{323}a^{13}-\frac{909}{323}a^{12}+\frac{2699}{323}a^{11}+\frac{110}{323}a^{10}-\frac{1613}{323}a^{9}+\frac{394}{323}a^{8}-\frac{1312}{323}a^{7}+\frac{1448}{323}a^{6}-\frac{622}{323}a^{5}-\frac{1779}{323}a^{4}+\frac{94}{17}a^{3}+\frac{233}{323}a^{2}-\frac{129}{323}a+\frac{74}{323}$, $\frac{139}{323}a^{14}+\frac{344}{323}a^{13}+\frac{341}{323}a^{12}-\frac{1096}{323}a^{11}-\frac{290}{323}a^{10}+\frac{905}{323}a^{9}+\frac{312}{323}a^{8}+\frac{317}{323}a^{7}-\frac{470}{323}a^{6}-\frac{122}{323}a^{5}+\frac{403}{323}a^{4}-\frac{33}{17}a^{3}-\frac{673}{323}a^{2}+\frac{252}{323}a+\frac{216}{323}$, $\frac{46}{323}a^{14}+\frac{237}{323}a^{13}-\frac{166}{323}a^{12}-\frac{695}{323}a^{11}+\frac{880}{323}a^{10}+\frac{339}{323}a^{9}-\frac{724}{323}a^{8}+\frac{486}{323}a^{7}-\frac{367}{323}a^{6}+\frac{192}{323}a^{5}-\frac{20}{323}a^{4}-\frac{47}{17}a^{3}+\frac{572}{323}a^{2}-\frac{63}{323}a-\frac{54}{323}$, $\frac{398}{323}a^{14}-\frac{899}{323}a^{13}-\frac{909}{323}a^{12}+\frac{2699}{323}a^{11}+\frac{110}{323}a^{10}-\frac{1613}{323}a^{9}+\frac{394}{323}a^{8}-\frac{1312}{323}a^{7}+\frac{1448}{323}a^{6}-\frac{622}{323}a^{5}-\frac{1779}{323}a^{4}+\frac{94}{17}a^{3}+\frac{233}{323}a^{2}+\frac{194}{323}a+\frac{74}{323}$, $\frac{53}{323}a^{14}+\frac{252}{323}a^{13}-\frac{107}{323}a^{12}-\frac{878}{323}a^{11}+\frac{719}{323}a^{10}+\frac{1170}{323}a^{9}-\frac{778}{323}a^{8}-\frac{718}{323}a^{7}-\frac{149}{323}a^{6}+\frac{474}{323}a^{5}+\frac{314}{323}a^{4}-\frac{39}{17}a^{3}-\frac{1}{323}a^{2}+\frac{763}{323}a+\frac{8}{323}$, $\frac{12}{323}a^{14}-\frac{118}{323}a^{13}+\frac{268}{323}a^{12}+\frac{83}{323}a^{11}-\frac{693}{323}a^{10}+\frac{375}{323}a^{9}-\frac{92}{323}a^{8}+\frac{126}{323}a^{7}+\frac{826}{323}a^{6}-\frac{991}{323}a^{5}+\frac{581}{323}a^{4}-\frac{4}{17}a^{3}-\frac{725}{323}a^{2}+\frac{199}{323}a-\frac{337}{323}$ 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:  \( 82.95238709015337 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 82.95238709015337 \cdot 1}{2\cdot\sqrt{12205980432301041}}\cr\approx \mathstrut & 0.184791399622152 \end{aligned}\]

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 - 3*x^14 + 7*x^12 - 6*x^11 + 4*x^9 - 6*x^8 + 6*x^7 - 7*x^6 + 6*x^4 - 5*x^3 + 3*x^2 + 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 - 3*x^14 + 7*x^12 - 6*x^11 + 4*x^9 - 6*x^8 + 6*x^7 - 7*x^6 + 6*x^4 - 5*x^3 + 3*x^2 + 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 - 3*x^14 + 7*x^12 - 6*x^11 + 4*x^9 - 6*x^8 + 6*x^7 - 7*x^6 + 6*x^4 - 5*x^3 + 3*x^2 + 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 = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^14 + 7*x^12 - 6*x^11 + 4*x^9 - 6*x^8 + 6*x^7 - 7*x^6 + 6*x^4 - 5*x^3 + 3*x^2 + 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

$D_5\wr C_3$ (as 15T50):

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); G, transitive_group_identification(G)
 
A solvable group of order 3000
The 32 conjugacy class representatives for $D_5\wr C_3$
Character table for $D_5\wr C_3$

Intermediate fields

\(\Q(\zeta_{9})^+\)

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 $15$ R $15$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ $15$ ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ $15$ ${\href{/padicField/37.5.0.1}{5} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ $15$ $15$ ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $15$

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
\(3\) Copy content Toggle raw display 3.5.3.20a2.1$x^{15} + 6 x^{11} + 9 x^{10} + 12 x^{7} + 36 x^{6} + 15 x^{5} + 8 x^{3} + 36 x^{2} + 30 x + 10$$3$$5$$20$$C_{15}$$$[2]^{5}$$
\(1871\) Copy content Toggle raw display $\Q_{1871}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1871}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1871}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1871}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1871}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1871}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1871}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Spectrum of ring of integers

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