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Properties

Label 14.0.118236562059083.1
Degree $14$
Signature $(0, 7)$
Discriminant $-1.182\times 10^{14}$
Root discriminant \(10.12\)
Ramified primes $13,71$
Class number $1$
Class group trivial
Galois group $C_2\wr D_7$ (as 14T38)

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Normalized defining polynomial

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

\( x^{14} - 3 x^{13} + 2 x^{12} + 3 x^{11} - 5 x^{10} - x^{9} + x^{8} + 2 x^{7} + 11 x^{6} - 13 x^{5} + \cdots + 1 \) 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:  $14$
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:  $(0, 7)$
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:   \(-118236562059083\) \(\medspace = -\,13\cdot 71^{7}\) 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:  \(10.12\)
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:  $13^{1/2}71^{3/4}\approx 88.18920705302763$
Ramified primes:   \(13\), \(71\) 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:  \(\Q(\sqrt{-923}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13823}a^{13}+\frac{6906}{13823}a^{12}-\frac{3440}{13823}a^{11}-\frac{5220}{13823}a^{10}-\frac{778}{13823}a^{9}+\frac{1944}{13823}a^{8}-\frac{4859}{13823}a^{7}+\frac{5238}{13823}a^{6}+\frac{739}{13823}a^{5}+\frac{5051}{13823}a^{4}-\frac{5710}{13823}a^{3}+\frac{444}{13823}a^{2}-\frac{1106}{13823}a+\frac{2765}{13823}$ 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:  $6$
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:   $\frac{1893}{13823}a^{13}-\frac{17323}{13823}a^{12}+\frac{26359}{13823}a^{11}+\frac{1985}{13823}a^{10}-\frac{48985}{13823}a^{9}+\frac{30720}{13823}a^{8}+\frac{35677}{13823}a^{7}+\frac{4443}{13823}a^{6}+\frac{30450}{13823}a^{5}-\frac{100734}{13823}a^{4}+\frac{83494}{13823}a^{3}-\frac{44180}{13823}a^{2}+\frac{21261}{13823}a-\frac{4772}{13823}$, $\frac{4547}{13823}a^{13}-\frac{18097}{13823}a^{12}+\frac{19779}{13823}a^{11}+\frac{12574}{13823}a^{10}-\frac{40347}{13823}a^{9}+\frac{6471}{13823}a^{8}+\frac{22927}{13823}a^{7}+\frac{13980}{13823}a^{6}+\frac{28890}{13823}a^{5}-\frac{117513}{13823}a^{4}+\frac{65339}{13823}a^{3}-\frac{26936}{13823}a^{2}+\frac{44059}{13823}a-\frac{6475}{13823}$, $\frac{909}{13823}a^{13}+\frac{1912}{13823}a^{12}-\frac{2962}{13823}a^{11}-\frac{3691}{13823}a^{10}+\frac{11594}{13823}a^{9}-\frac{2248}{13823}a^{8}-\frac{21117}{13823}a^{7}-\frac{7593}{13823}a^{6}-\frac{5576}{13823}a^{5}+\frac{15946}{13823}a^{4}+\frac{20881}{13823}a^{3}-\frac{11094}{13823}a^{2}+\frac{17548}{13823}a-\frac{2401}{13823}$, $\frac{9289}{13823}a^{13}-\frac{30355}{13823}a^{12}+\frac{18439}{13823}a^{11}+\frac{43973}{13823}a^{10}-\frac{66528}{13823}a^{9}-\frac{22668}{13823}a^{8}+\frac{52136}{13823}a^{7}+\frac{26468}{13823}a^{6}+\frac{77478}{13823}a^{5}-\frac{148576}{13823}a^{4}+\frac{26307}{13823}a^{3}-\frac{8761}{13823}a^{2}+\frac{10678}{13823}a+\frac{14774}{13823}$, $\frac{4742}{13823}a^{13}-\frac{12258}{13823}a^{12}-\frac{1340}{13823}a^{11}+\frac{31399}{13823}a^{10}-\frac{26181}{13823}a^{9}-\frac{29139}{13823}a^{8}+\frac{29209}{13823}a^{7}+\frac{12488}{13823}a^{6}+\frac{48588}{13823}a^{5}-\frac{31063}{13823}a^{4}-\frac{39032}{13823}a^{3}+\frac{18175}{13823}a^{2}-\frac{33381}{13823}a+\frac{21249}{13823}$, $\frac{3966}{13823}a^{13}-\frac{7990}{13823}a^{12}+\frac{261}{13823}a^{11}+\frac{18157}{13823}a^{10}-\frac{16842}{13823}a^{9}-\frac{17153}{13823}a^{8}+\frac{12291}{13823}a^{7}-\frac{2061}{13823}a^{6}+\frac{14221}{13823}a^{5}-\frac{24907}{13823}a^{4}+\frac{10037}{13823}a^{3}+\frac{33029}{13823}a^{2}-\frac{4505}{13823}a+\frac{4351}{13823}$ 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:  \( 12.9746502268 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 12.9746502268 \cdot 1}{2\cdot\sqrt{118236562059083}}\cr\approx \mathstrut & 0.230647549616 \end{aligned}\]

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^14 - 3*x^13 + 2*x^12 + 3*x^11 - 5*x^10 - x^9 + x^8 + 2*x^7 + 11*x^6 - 13*x^5 + 6*x^4 - 8*x^3 + 4*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^14 - 3*x^13 + 2*x^12 + 3*x^11 - 5*x^10 - x^9 + x^8 + 2*x^7 + 11*x^6 - 13*x^5 + 6*x^4 - 8*x^3 + 4*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^14 - 3*x^13 + 2*x^12 + 3*x^11 - 5*x^10 - x^9 + x^8 + 2*x^7 + 11*x^6 - 13*x^5 + 6*x^4 - 8*x^3 + 4*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 = polynomial_ring(QQ); K, a = number_field(x^14 - 3*x^13 + 2*x^12 + 3*x^11 - 5*x^10 - x^9 + x^8 + 2*x^7 + 11*x^6 - 13*x^5 + 6*x^4 - 8*x^3 + 4*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

$C_2\wr D_7$ (as 14T38):

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 1792
The 40 conjugacy class representatives for $C_2\wr D_7$
Character table for $C_2\wr D_7$

Intermediate fields

7.1.357911.1

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 14 siblings: data not computed
Degree 28 siblings: data not computed
Degree 32 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 ${\href{/padicField/2.14.0.1}{14} }$ ${\href{/padicField/3.7.0.1}{7} }^{2}$ ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ R ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.14.0.1}{14} }$ ${\href{/padicField/23.2.0.1}{2} }^{7}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ ${\href{/padicField/37.14.0.1}{14} }$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{5}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(13\) Copy content Toggle raw display 13.1.2.1a1.1$x^{2} + 13$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(71\) Copy content Toggle raw display $\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$$$[\ ]_{2}$$
71.2.2.2a1.2$x^{4} + 138 x^{3} + 4775 x^{2} + 966 x + 120$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
71.1.4.3a1.1$x^{4} + 71$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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