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Properties

Label 14.4.1524052425039147.1
Degree $14$
Signature $[4, 5]$
Discriminant $-1.524\times 10^{15}$
Root discriminant \(12.15\)
Ramified primes $3,2741$
Class number $1$
Class group trivial
Galois group $C_2^4:\GL(3,2)$ (as 14T43)

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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 - 6*x^13 + 20*x^12 - 46*x^11 + 79*x^10 - 108*x^9 + 113*x^8 - 92*x^7 + 46*x^6 - 5*x^5 - 21*x^4 + 19*x^3 - 10*x^2 + 1)
 
Copy content gp:K = bnfinit(y^14 - 6*y^13 + 20*y^12 - 46*y^11 + 79*y^10 - 108*y^9 + 113*y^8 - 92*y^7 + 46*y^6 - 5*y^5 - 21*y^4 + 19*y^3 - 10*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 6*x^13 + 20*x^12 - 46*x^11 + 79*x^10 - 108*x^9 + 113*x^8 - 92*x^7 + 46*x^6 - 5*x^5 - 21*x^4 + 19*x^3 - 10*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 6*x^13 + 20*x^12 - 46*x^11 + 79*x^10 - 108*x^9 + 113*x^8 - 92*x^7 + 46*x^6 - 5*x^5 - 21*x^4 + 19*x^3 - 10*x^2 + 1)
 

\( x^{14} - 6 x^{13} + 20 x^{12} - 46 x^{11} + 79 x^{10} - 108 x^{9} + 113 x^{8} - 92 x^{7} + 46 x^{6} + \cdots + 1 \) 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:  $14$
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:  $[4, 5]$
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:   \(-1524052425039147\) \(\medspace = -\,3^{3}\cdot 2741^{4}\) 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:  \(12.15\)
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:  $3^{1/2}2741^{1/2}\approx 90.68075870878012$
Ramified primes:   \(3\), \(2741\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3183}a^{13}-\frac{92}{3183}a^{12}+\frac{522}{1061}a^{11}-\frac{1036}{3183}a^{10}+\frac{17}{1061}a^{9}-\frac{437}{1061}a^{8}+\frac{1454}{3183}a^{7}-\frac{333}{1061}a^{6}+\frac{19}{3183}a^{5}+\frac{1544}{3183}a^{4}+\frac{881}{3183}a^{3}+\frac{215}{1061}a^{2}-\frac{1369}{3183}a-\frac{37}{3183}$ 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:   $\frac{351}{1061}a^{13}-\frac{1523}{1061}a^{12}+\frac{4312}{1061}a^{11}-\frac{9262}{1061}a^{10}+\frac{15779}{1061}a^{9}-\frac{23029}{1061}a^{8}+\frac{24416}{1061}a^{7}-\frac{23861}{1061}a^{6}+\frac{14096}{1061}a^{5}-\frac{6593}{1061}a^{4}-\frac{2703}{1061}a^{3}+\frac{3585}{1061}a^{2}-\frac{3069}{1061}a-\frac{255}{1061}$, $\frac{1652}{3183}a^{13}-\frac{8749}{3183}a^{12}+\frac{8239}{1061}a^{11}-\frac{46763}{3183}a^{10}+\frac{20657}{1061}a^{9}-\frac{20603}{1061}a^{8}+\frac{33856}{3183}a^{7}+\frac{543}{1061}a^{6}-\frac{32272}{3183}a^{5}+\frac{36118}{3183}a^{4}-\frac{18317}{3183}a^{3}+\frac{806}{1061}a^{2}+\frac{7891}{3183}a-\frac{647}{3183}$, $\frac{254}{1061}a^{13}-\frac{1087}{1061}a^{12}+\frac{2011}{1061}a^{11}-\frac{1077}{1061}a^{10}-\frac{4022}{1061}a^{9}+\frac{11831}{1061}a^{8}-\frac{20071}{1061}a^{7}+\frac{22114}{1061}a^{6}-\frac{16394}{1061}a^{5}+\frac{4911}{1061}a^{4}+\frac{5208}{1061}a^{3}-\frac{6991}{1061}a^{2}+\frac{4526}{1061}a+\frac{151}{1061}$, $\frac{421}{3183}a^{13}-\frac{3719}{3183}a^{12}+\frac{4379}{1061}a^{11}-\frac{28732}{3183}a^{10}+\frac{14584}{1061}a^{9}-\frac{16339}{1061}a^{8}+\frac{42377}{3183}a^{7}-\frac{6507}{1061}a^{6}+\frac{1633}{3183}a^{5}+\frac{10241}{3183}a^{4}-\frac{4693}{3183}a^{3}+\frac{330}{1061}a^{2}+\frac{6140}{3183}a+\frac{338}{3183}$, $\frac{2533}{3183}a^{13}-\frac{13409}{3183}a^{12}+\frac{12952}{1061}a^{11}-\frac{77788}{3183}a^{10}+\frac{38817}{1061}a^{9}-\frac{48043}{1061}a^{8}+\frac{133937}{3183}a^{7}-\frac{33946}{1061}a^{6}+\frac{48127}{3183}a^{5}-\frac{13687}{3183}a^{4}-\frac{9259}{3183}a^{3}+\frac{2424}{1061}a^{2}-\frac{7756}{3183}a+\frac{1769}{3183}$, $\frac{518}{3183}a^{13}-\frac{3094}{3183}a^{12}+\frac{3024}{1061}a^{11}-\frac{17819}{3183}a^{10}+\frac{8806}{1061}a^{9}-\frac{10983}{1061}a^{8}+\frac{33814}{3183}a^{7}-\frac{9100}{1061}a^{6}+\frac{19391}{3183}a^{5}-\frac{11873}{3183}a^{4}+\frac{4372}{3183}a^{3}-\frac{35}{1061}a^{2}-\frac{2516}{3183}a+\frac{3115}{3183}$, $\frac{1190}{3183}a^{13}-\frac{7624}{3183}a^{12}+\frac{8983}{1061}a^{11}-\frac{64679}{3183}a^{10}+\frac{38267}{1061}a^{9}-\frac{53190}{1061}a^{8}+\frac{170590}{3183}a^{7}-\frac{47201}{1061}a^{6}+\frac{73538}{3183}a^{5}-\frac{11963}{3183}a^{4}-\frac{30647}{3183}a^{3}+\frac{8637}{1061}a^{2}-\frac{15329}{3183}a-\frac{2651}{3183}$, $a$ 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:  \( 121.507606686 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 121.507606686 \cdot 1}{2\cdot\sqrt{1524052425039147}}\cr\approx \mathstrut & 0.243833202297 \end{aligned}\]

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 - 6*x^13 + 20*x^12 - 46*x^11 + 79*x^10 - 108*x^9 + 113*x^8 - 92*x^7 + 46*x^6 - 5*x^5 - 21*x^4 + 19*x^3 - 10*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 - 6*x^13 + 20*x^12 - 46*x^11 + 79*x^10 - 108*x^9 + 113*x^8 - 92*x^7 + 46*x^6 - 5*x^5 - 21*x^4 + 19*x^3 - 10*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 - 6*x^13 + 20*x^12 - 46*x^11 + 79*x^10 - 108*x^9 + 113*x^8 - 92*x^7 + 46*x^6 - 5*x^5 - 21*x^4 + 19*x^3 - 10*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^14 - 6*x^13 + 20*x^12 - 46*x^11 + 79*x^10 - 108*x^9 + 113*x^8 - 92*x^7 + 46*x^6 - 5*x^5 - 21*x^4 + 19*x^3 - 10*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

$C_2^4:\GL(3,2)$ (as 14T43):

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 non-solvable group of order 2688
The 22 conjugacy class representatives for $C_2^4:\GL(3,2)$
Character table for $C_2^4:\GL(3,2)$

Intermediate fields

7.3.7513081.2

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 14 sibling: data not computed
Degree 16 siblings: data not computed
Degree 28 siblings: data not computed
Degree 42 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 ${\href{/padicField/2.14.0.1}{14} }$ R ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.7.0.1}{7} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.14.0.1}{14} }$

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 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(2741\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
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$$$[\ ]_{2}$$
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)