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Properties

Label 12.4.144215816802121.1
Degree $12$
Signature $(4, 4)$
Discriminant $1.442\times 10^{14}$
Root discriminant \(15.13\)
Ramified prime $229$
Class number $1$
Class group trivial
Galois group $S_4$ (as 12T9)

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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^12 - 2*x^10 - 6*x^9 - 6*x^8 - 5*x^7 + 8*x^6 + 32*x^5 - 4*x^4 - 52*x^3 + x^2 + 44*x + 16)
 
Copy content gp:K = bnfinit(y^12 - 2*y^10 - 6*y^9 - 6*y^8 - 5*y^7 + 8*y^6 + 32*y^5 - 4*y^4 - 52*y^3 + y^2 + 44*y + 16, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^10 - 6*x^9 - 6*x^8 - 5*x^7 + 8*x^6 + 32*x^5 - 4*x^4 - 52*x^3 + x^2 + 44*x + 16);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^10 - 6*x^9 - 6*x^8 - 5*x^7 + 8*x^6 + 32*x^5 - 4*x^4 - 52*x^3 + x^2 + 44*x + 16)
 

\( x^{12} - 2x^{10} - 6x^{9} - 6x^{8} - 5x^{7} + 8x^{6} + 32x^{5} - 4x^{4} - 52x^{3} + x^{2} + 44x + 16 \) 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:  $12$
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:  $(4, 4)$
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:   \(144215816802121\) \(\medspace = 229^{6}\) 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:  \(15.13\)
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:  $229^{1/2}\approx 15.132745950421556$
Ramified primes:   \(229\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{21}a^{10}+\frac{5}{21}a^{9}-\frac{8}{21}a^{8}+\frac{3}{7}a^{7}-\frac{1}{3}a^{6}-\frac{4}{21}a^{5}-\frac{5}{21}a^{4}+\frac{5}{21}a^{3}+\frac{8}{21}a^{2}+\frac{1}{21}a+\frac{10}{21}$, $\frac{1}{16332372}a^{11}+\frac{10663}{4083093}a^{10}-\frac{216641}{1166598}a^{9}-\frac{3843809}{8166186}a^{8}+\frac{2875057}{8166186}a^{7}+\frac{514575}{1814708}a^{6}+\frac{1292285}{4083093}a^{5}+\frac{2034019}{4083093}a^{4}+\frac{102013}{453677}a^{3}-\frac{835393}{4083093}a^{2}-\frac{483761}{5444124}a+\frac{1641749}{4083093}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  $C_{2}$, which has order $2$
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:  $7$
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:   $\frac{921827}{16332372}a^{11}-\frac{17002}{583299}a^{10}-\frac{1167421}{8166186}a^{9}-\frac{298915}{1166598}a^{8}-\frac{1332211}{8166186}a^{7}+\frac{554383}{5444124}a^{6}+\frac{2545243}{4083093}a^{5}+\frac{7941131}{4083093}a^{4}-\frac{2084867}{1361031}a^{3}-\frac{12594551}{4083093}a^{2}+\frac{9392161}{5444124}a+\frac{10912705}{4083093}$, $\frac{1282649}{8166186}a^{11}-\frac{88190}{583299}a^{10}-\frac{941650}{4083093}a^{9}-\frac{412072}{583299}a^{8}-\frac{787681}{4083093}a^{7}-\frac{171319}{907354}a^{6}+\frac{7254074}{4083093}a^{5}+\frac{15581773}{4083093}a^{4}-\frac{2184878}{453677}a^{3}-\frac{22380961}{4083093}a^{2}+\frac{13635785}{2722062}a+\frac{14904545}{4083093}$, $\frac{457469}{16332372}a^{11}-\frac{137590}{4083093}a^{10}-\frac{292351}{8166186}a^{9}-\frac{961237}{8166186}a^{8}+\frac{33779}{8166186}a^{7}+\frac{3623}{1814708}a^{6}+\frac{957175}{4083093}a^{5}+\frac{2824244}{4083093}a^{4}-\frac{412429}{453677}a^{3}-\frac{4061291}{4083093}a^{2}+\frac{4979279}{5444124}a+\frac{4563562}{4083093}$, $\frac{1631773}{16332372}a^{11}-\frac{602537}{4083093}a^{10}-\frac{917873}{8166186}a^{9}-\frac{3374501}{8166186}a^{8}+\frac{218143}{1166598}a^{7}-\frac{103331}{5444124}a^{6}+\frac{6863414}{4083093}a^{5}+\frac{7424755}{4083093}a^{4}-\frac{5520947}{1361031}a^{3}-\frac{11506006}{4083093}a^{2}+\frac{7774145}{1814708}a+\frac{950525}{583299}$, $\frac{976741}{5444124}a^{11}-\frac{268039}{1361031}a^{10}-\frac{584419}{2722062}a^{9}-\frac{783819}{907354}a^{8}-\frac{48113}{2722062}a^{7}-\frac{2430853}{5444124}a^{6}+\frac{483010}{194433}a^{5}+\frac{5153681}{1361031}a^{4}-\frac{908050}{194433}a^{3}-\frac{7918097}{1361031}a^{2}+\frac{4654639}{777732}a+\frac{1775350}{453677}$, $\frac{349793}{16332372}a^{11}+\frac{34520}{4083093}a^{10}-\frac{537931}{8166186}a^{9}-\frac{1010659}{8166186}a^{8}-\frac{1492081}{8166186}a^{7}-\frac{313555}{5444124}a^{6}+\frac{798295}{4083093}a^{5}+\frac{4125188}{4083093}a^{4}+\frac{610196}{1361031}a^{3}-\frac{458897}{583299}a^{2}+\frac{300793}{1814708}a+\frac{2582044}{4083093}$, $\frac{823507}{8166186}a^{11}-\frac{244876}{4083093}a^{10}+\frac{54967}{4083093}a^{9}-\frac{2491151}{4083093}a^{8}-\frac{1491566}{4083093}a^{7}-\frac{3569329}{2722062}a^{6}+\frac{1072075}{4083093}a^{5}+\frac{3233288}{4083093}a^{4}-\frac{2128424}{1361031}a^{3}-\frac{1056722}{583299}a^{2}-\frac{2000113}{2722062}a-\frac{1393205}{4083093}$ 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:  \( 328.61155497 \)
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:  \( 3 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 328.61155497 \cdot 1}{2\cdot\sqrt{144215816802121}}\cr\approx \mathstrut & 0.34118179038 \end{aligned}\]

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^12 - 2*x^10 - 6*x^9 - 6*x^8 - 5*x^7 + 8*x^6 + 32*x^5 - 4*x^4 - 52*x^3 + x^2 + 44*x + 16) 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^12 - 2*x^10 - 6*x^9 - 6*x^8 - 5*x^7 + 8*x^6 + 32*x^5 - 4*x^4 - 52*x^3 + x^2 + 44*x + 16, 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^12 - 2*x^10 - 6*x^9 - 6*x^8 - 5*x^7 + 8*x^6 + 32*x^5 - 4*x^4 - 52*x^3 + x^2 + 44*x + 16); 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^12 - 2*x^10 - 6*x^9 - 6*x^8 - 5*x^7 + 8*x^6 + 32*x^5 - 4*x^4 - 52*x^3 + x^2 + 44*x + 16); 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

$S_4$ (as 12T9):

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 24
The 5 conjugacy class representatives for $S_4$
Character table for $S_4$

Intermediate fields

\(\Q(\sqrt{229}) \), 3.3.229.1 x3, 6.2.12008989.1, 6.2.52441.1, 6.6.12008989.1

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

Galois closure: deg 24
Degree 4 sibling: 4.0.229.1
Degree 6 siblings: 6.2.12008989.1, 6.2.52441.1
Degree 8 sibling: 8.0.2750058481.1
Degree 12 sibling: 12.0.629763392149.1
Minimal sibling: 4.0.229.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ ${\href{/padicField/3.3.0.1}{3} }^{4}$ ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(229\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$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)