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Properties

Label 14.4.370...103.1
Degree $14$
Signature $[4, 5]$
Discriminant $-3.704\times 10^{18}$
Root discriminant \(21.20\)
Ramified prime $3704349534869766103$
Class number $1$
Class group trivial
Galois group $S_{14}$ (as 14T63)

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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^14 - 3*x^13 - 9*x^12 + 31*x^11 + 27*x^10 - 119*x^9 - 27*x^8 + 211*x^7 - 7*x^6 - 175*x^5 + 24*x^4 + 59*x^3 - 10*x^2 - 5*x + 3)
 
Copy content gp:K = bnfinit(y^14 - 3*y^13 - 9*y^12 + 31*y^11 + 27*y^10 - 119*y^9 - 27*y^8 + 211*y^7 - 7*y^6 - 175*y^5 + 24*y^4 + 59*y^3 - 10*y^2 - 5*y + 3, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 - 9*x^12 + 31*x^11 + 27*x^10 - 119*x^9 - 27*x^8 + 211*x^7 - 7*x^6 - 175*x^5 + 24*x^4 + 59*x^3 - 10*x^2 - 5*x + 3);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 3*x^13 - 9*x^12 + 31*x^11 + 27*x^10 - 119*x^9 - 27*x^8 + 211*x^7 - 7*x^6 - 175*x^5 + 24*x^4 + 59*x^3 - 10*x^2 - 5*x + 3)
 

\( x^{14} - 3 x^{13} - 9 x^{12} + 31 x^{11} + 27 x^{10} - 119 x^{9} - 27 x^{8} + 211 x^{7} - 7 x^{6} + \cdots + 3 \) 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:   \(-3704349534869766103\) 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:  \(21.20\)
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:  $3704349534869766103^{1/2}\approx 1924668681.8436482$
Ramified primes:   \(3704349534869766103\) 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:  $\Q(\sqrt{-37043\!\cdots\!66103}$)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{59}a^{13}-\frac{16}{59}a^{12}+\frac{22}{59}a^{11}-\frac{19}{59}a^{10}-\frac{21}{59}a^{9}-\frac{23}{59}a^{8}-\frac{23}{59}a^{7}-\frac{21}{59}a^{6}-\frac{29}{59}a^{5}+\frac{25}{59}a^{4}-\frac{6}{59}a^{3}+\frac{19}{59}a^{2}-\frac{21}{59}a-\frac{27}{59}$ 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:   $\frac{13}{59}a^{13}-\frac{31}{59}a^{12}-\frac{127}{59}a^{11}+\frac{284}{59}a^{10}+\frac{494}{59}a^{9}-\frac{889}{59}a^{8}-\frac{1066}{59}a^{7}+\frac{1084}{59}a^{6}+\frac{1393}{59}a^{5}-\frac{442}{59}a^{4}-\frac{845}{59}a^{3}+\frac{129}{59}a^{2}+\frac{140}{59}a-\frac{56}{59}$, $a-1$, $a^{13}-3a^{12}-8a^{11}+29a^{10}+17a^{9}-100a^{8}+9a^{7}+147a^{6}-62a^{5}-83a^{4}+54a^{3}+6a^{2}-9a+2$, $\frac{35}{59}a^{13}-\frac{88}{59}a^{12}-\frac{351}{59}a^{11}+\frac{928}{59}a^{10}+\frac{1271}{59}a^{9}-\frac{3637}{59}a^{8}-\frac{1985}{59}a^{7}+\frac{6522}{59}a^{6}+\frac{1227}{59}a^{5}-\frac{5261}{59}a^{4}-\frac{269}{59}a^{3}+\frac{1550}{59}a^{2}+\frac{209}{59}a-\frac{119}{59}$, $\frac{17}{59}a^{13}-\frac{36}{59}a^{12}-\frac{157}{59}a^{11}+\frac{326}{59}a^{10}+\frac{528}{59}a^{9}-\frac{1040}{59}a^{8}-\frac{863}{59}a^{7}+\frac{1472}{59}a^{6}+\frac{864}{59}a^{5}-\frac{1109}{59}a^{4}-\frac{515}{59}a^{3}+\frac{500}{59}a^{2}+\frac{56}{59}a-\frac{46}{59}$, $\frac{2}{59}a^{13}-\frac{32}{59}a^{12}+\frac{44}{59}a^{11}+\frac{257}{59}a^{10}-\frac{455}{59}a^{9}-\frac{636}{59}a^{8}+\frac{1311}{59}a^{7}+\frac{430}{59}a^{6}-\frac{1356}{59}a^{5}+\frac{109}{59}a^{4}+\frac{519}{59}a^{3}-\frac{80}{59}a^{2}-\frac{101}{59}a+\frac{5}{59}$, $\frac{26}{59}a^{13}-\frac{62}{59}a^{12}-\frac{254}{59}a^{11}+\frac{627}{59}a^{10}+\frac{870}{59}a^{9}-\frac{2309}{59}a^{8}-\frac{1129}{59}a^{7}+\frac{3761}{59}a^{6}+\frac{13}{59}a^{5}-\frac{2595}{59}a^{4}+\frac{1024}{59}a^{3}+\frac{553}{59}a^{2}-\frac{428}{59}a+\frac{65}{59}$, $\frac{41}{59}a^{13}-\frac{66}{59}a^{12}-\frac{455}{59}a^{11}+\frac{637}{59}a^{10}+\frac{1912}{59}a^{9}-\frac{2182}{59}a^{8}-\frac{3775}{59}a^{7}+\frac{3210}{59}a^{6}+\frac{3472}{59}a^{5}-\frac{1984}{59}a^{4}-\frac{1131}{59}a^{3}+\frac{484}{59}a^{2}-\frac{35}{59}a-\frac{104}{59}$ 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:  \( 11913.6860087 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 11913.6860087 \cdot 1}{2\cdot\sqrt{3704349534869766103}}\cr\approx \mathstrut & 0.484930498781 \end{aligned}\]

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 - 9*x^12 + 31*x^11 + 27*x^10 - 119*x^9 - 27*x^8 + 211*x^7 - 7*x^6 - 175*x^5 + 24*x^4 + 59*x^3 - 10*x^2 - 5*x + 3) 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 - 9*x^12 + 31*x^11 + 27*x^10 - 119*x^9 - 27*x^8 + 211*x^7 - 7*x^6 - 175*x^5 + 24*x^4 + 59*x^3 - 10*x^2 - 5*x + 3, 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 - 9*x^12 + 31*x^11 + 27*x^10 - 119*x^9 - 27*x^8 + 211*x^7 - 7*x^6 - 175*x^5 + 24*x^4 + 59*x^3 - 10*x^2 - 5*x + 3); 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 - 3*x^13 - 9*x^12 + 31*x^11 + 27*x^10 - 119*x^9 - 27*x^8 + 211*x^7 - 7*x^6 - 175*x^5 + 24*x^4 + 59*x^3 - 10*x^2 - 5*x + 3); 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

$S_{14}$ (as 14T63):

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 87178291200
The 135 conjugacy class representatives for $S_{14}$
Character table for $S_{14}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 28 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 ${\href{/padicField/2.7.0.1}{7} }^{2}$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.7.0.1}{7} }^{2}$ ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.14.0.1}{14} }$ ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(3704349534869766103\) Copy content Toggle raw display $\Q_{3704349534869766103}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3704349534869766103}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $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)