Properties

Label 15.3.124...000.1
Degree $15$
Signature $(3, 6)$
Discriminant $1.240\times 10^{63}$
Root discriminant \($16\,078$.06\)
Ramified primes $2,5,31,43,71$
Class number not computed
Class group not computed
Galois group $F_5 \times S_3$ (as 15T11)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - 5612*x^13 + 76028*x^12 + 15049768*x^11 - 156320606*x^10 - 20634964056*x^9 + 261790345932*x^8 + 22994910881888*x^7 - 57978950329640*x^6 - 11909676364684660*x^5 + 40308498022610840*x^4 + 7263121818166906744*x^3 + 128646924961463259264*x^2 + 1443546768447897363040*x + 11039634591408758044088)
 
Copy content gp:K = bnfinit(y^15 - 6*y^14 - 5612*y^13 + 76028*y^12 + 15049768*y^11 - 156320606*y^10 - 20634964056*y^9 + 261790345932*y^8 + 22994910881888*y^7 - 57978950329640*y^6 - 11909676364684660*y^5 + 40308498022610840*y^4 + 7263121818166906744*y^3 + 128646924961463259264*y^2 + 1443546768447897363040*y + 11039634591408758044088, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 6*x^14 - 5612*x^13 + 76028*x^12 + 15049768*x^11 - 156320606*x^10 - 20634964056*x^9 + 261790345932*x^8 + 22994910881888*x^7 - 57978950329640*x^6 - 11909676364684660*x^5 + 40308498022610840*x^4 + 7263121818166906744*x^3 + 128646924961463259264*x^2 + 1443546768447897363040*x + 11039634591408758044088);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 6*x^14 - 5612*x^13 + 76028*x^12 + 15049768*x^11 - 156320606*x^10 - 20634964056*x^9 + 261790345932*x^8 + 22994910881888*x^7 - 57978950329640*x^6 - 11909676364684660*x^5 + 40308498022610840*x^4 + 7263121818166906744*x^3 + 128646924961463259264*x^2 + 1443546768447897363040*x + 11039634591408758044088)
 

\( x^{15} - 6 x^{14} - 5612 x^{13} + 76028 x^{12} + 15049768 x^{11} - 156320606 x^{10} + \cdots + 11\!\cdots\!88 \) 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:   \(1240233977923368391582786353067528369617276883372654592000000000\) \(\medspace = 2^{23}\cdot 5^{9}\cdot 31^{13}\cdot 43^{13}\cdot 71^{5}\) 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:  \($16\,078$.06\)
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:  $2^{19/10}5^{3/4}31^{9/10}43^{9/10}71^{1/2}\approx 68259.13867415124$
Ramified primes:   \(2\), \(5\), \(31\), \(43\), \(71\) 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:  $\Q(\sqrt{946430}$)
$\Aut(K/\Q)$:   $C_1$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{5332}a^{10}-\frac{311}{5332}a^{9}+\frac{23}{2666}a^{8}-\frac{290}{1333}a^{7}-\frac{543}{2666}a^{6}-\frac{95}{2666}a^{5}-\frac{1159}{2666}a^{4}+\frac{236}{1333}a^{3}+\frac{411}{1333}a^{2}+\frac{291}{1333}a-\frac{102}{1333}$, $\frac{1}{5332}a^{11}+\frac{317}{2666}a^{9}-\frac{46}{1333}a^{8}+\frac{365}{2666}a^{7}+\frac{323}{2666}a^{6}-\frac{45}{2666}a^{5}+\frac{633}{1333}a^{4}+\frac{492}{1333}a^{3}+\frac{144}{1333}a^{2}-\frac{245}{1333}a+\frac{270}{1333}$, $\frac{1}{3700808123944}a^{12}-\frac{109376117}{1850404061972}a^{11}+\frac{161104669}{1850404061972}a^{10}-\frac{14506649691}{925202030986}a^{9}-\frac{158323567973}{925202030986}a^{8}-\frac{211982423551}{925202030986}a^{7}+\frac{74740501870}{462601015493}a^{6}-\frac{1913931031}{14922613403}a^{5}-\frac{17124591842}{462601015493}a^{4}-\frac{110803501044}{462601015493}a^{3}+\frac{198641250179}{925202030986}a^{2}-\frac{228063455745}{462601015493}a-\frac{138267801822}{462601015493}$, $\frac{1}{3700808123944}a^{13}-\frac{30823629}{925202030986}a^{11}+\frac{135474491}{1850404061972}a^{10}+\frac{30738242653}{1850404061972}a^{9}+\frac{73714535371}{462601015493}a^{8}-\frac{23831971440}{462601015493}a^{7}-\frac{83995237550}{462601015493}a^{6}+\frac{28828822357}{925202030986}a^{5}+\frac{58868766379}{925202030986}a^{4}+\frac{122718903969}{925202030986}a^{3}-\frac{158112407439}{462601015493}a^{2}+\frac{164463921698}{462601015493}a+\frac{12563938463}{462601015493}$, $\frac{1}{30\cdots 12}a^{14}+\frac{10\cdots 85}{15\cdots 56}a^{13}-\frac{37\cdots 11}{30\cdots 12}a^{12}-\frac{27\cdots 25}{37\cdots 89}a^{11}-\frac{54\cdots 25}{15\cdots 56}a^{10}-\frac{83\cdots 07}{15\cdots 56}a^{9}+\frac{25\cdots 89}{43\cdots 42}a^{8}+\frac{87\cdots 57}{37\cdots 89}a^{7}-\frac{16\cdots 75}{75\cdots 78}a^{6}-\frac{17\cdots 27}{24\cdots 26}a^{5}-\frac{16\cdots 96}{37\cdots 89}a^{4}-\frac{13\cdots 87}{37\cdots 89}a^{3}-\frac{14\cdots 31}{75\cdots 78}a^{2}+\frac{21\cdots 63}{37\cdots 89}a+\frac{11\cdots 94}{37\cdots 89}$ 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:  not computed
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:  not computed
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:  not computed
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:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot R \cdot h}{2\cdot\sqrt{1240233977923368391582786353067528369617276883372654592000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 - 6*x^14 - 5612*x^13 + 76028*x^12 + 15049768*x^11 - 156320606*x^10 - 20634964056*x^9 + 261790345932*x^8 + 22994910881888*x^7 - 57978950329640*x^6 - 11909676364684660*x^5 + 40308498022610840*x^4 + 7263121818166906744*x^3 + 128646924961463259264*x^2 + 1443546768447897363040*x + 11039634591408758044088) 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 - 6*x^14 - 5612*x^13 + 76028*x^12 + 15049768*x^11 - 156320606*x^10 - 20634964056*x^9 + 261790345932*x^8 + 22994910881888*x^7 - 57978950329640*x^6 - 11909676364684660*x^5 + 40308498022610840*x^4 + 7263121818166906744*x^3 + 128646924961463259264*x^2 + 1443546768447897363040*x + 11039634591408758044088, 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 - 6*x^14 - 5612*x^13 + 76028*x^12 + 15049768*x^11 - 156320606*x^10 - 20634964056*x^9 + 261790345932*x^8 + 22994910881888*x^7 - 57978950329640*x^6 - 11909676364684660*x^5 + 40308498022610840*x^4 + 7263121818166906744*x^3 + 128646924961463259264*x^2 + 1443546768447897363040*x + 11039634591408758044088); 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 - 6*x^14 - 5612*x^13 + 76028*x^12 + 15049768*x^11 - 156320606*x^10 - 20634964056*x^9 + 261790345932*x^8 + 22994910881888*x^7 - 57978950329640*x^6 - 11909676364684660*x^5 + 40308498022610840*x^4 + 7263121818166906744*x^3 + 128646924961463259264*x^2 + 1443546768447897363040*x + 11039634591408758044088); 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

$S_3\times F_5$ (as 15T11):

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 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.3.757144.1, 5.1.6314669036642000.1

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 30 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 R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ R ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ $15$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ R ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ R ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\) Copy content Toggle raw display 2.1.5.4a1.1$x^{5} + 2$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
2.1.10.19a1.2$x^{10} + 10$$10$$1$$19$$F_{5}\times C_2$$$[3]_{5}^{4}$$
\(5\) Copy content Toggle raw display 5.3.1.0a1.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
5.3.4.9a1.3$x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 486 x^{2} + 324 x + 86$$4$$3$$9$$C_{12}$$$[\ ]_{4}^{3}$$
\(31\) Copy content Toggle raw display 31.1.5.4a1.4$x^{5} + 589$$5$$1$$4$$C_5$$$[\ ]_{5}$$
31.1.10.9a1.6$x^{10} + 589$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$
\(43\) Copy content Toggle raw display 43.1.5.4a1.1$x^{5} + 43$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
43.1.10.9a1.2$x^{10} + 129$$10$$1$$9$$F_{5}\times C_2$$$[\ ]_{10}^{4}$$
\(71\) Copy content Toggle raw display $\Q_{71}$$x + 64$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{71}$$x + 64$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{71}$$x + 64$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{71}$$x + 64$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{71}$$x + 64$$1$$1$$0$Trivial$$[\ ]$$
71.1.2.1a1.1$x^{2} + 71$$2$$1$$1$$C_2$$$[\ ]_{2}$$
71.1.2.1a1.1$x^{2} + 71$$2$$1$$1$$C_2$$$[\ ]_{2}$$
71.1.2.1a1.1$x^{2} + 71$$2$$1$$1$$C_2$$$[\ ]_{2}$$
71.1.2.1a1.1$x^{2} + 71$$2$$1$$1$$C_2$$$[\ ]_{2}$$
71.1.2.1a1.1$x^{2} + 71$$2$$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)