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Properties

Label 12.2.3189382584467456.3
Degree $12$
Signature $[2, 5]$
Discriminant $-3.189\times 10^{15}$
Root discriminant \(19.59\)
Ramified primes $2,13$
Class number $1$
Class group trivial
Galois group $S_4$ (as 12T8)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 + 8*x^10 - 12*x^9 + 10*x^8 - 10*x^6 - 16*x^4 + 12*x^3 + 8*x^2 + 1)
 
Copy content gp:K = bnfinit(y^12 + 8*y^10 - 12*y^9 + 10*y^8 - 10*y^6 - 16*y^4 + 12*y^3 + 8*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 8*x^10 - 12*x^9 + 10*x^8 - 10*x^6 - 16*x^4 + 12*x^3 + 8*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 + 8*x^10 - 12*x^9 + 10*x^8 - 10*x^6 - 16*x^4 + 12*x^3 + 8*x^2 + 1)
 

\( x^{12} + 8x^{10} - 12x^{9} + 10x^{8} - 10x^{6} - 16x^{4} + 12x^{3} + 8x^{2} + 1 \) 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:  $12$
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:  $[2, 5]$
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:   \(-3189382584467456\) \(\medspace = -\,2^{33}\cdot 13^{5}\) 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:  \(19.59\)
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:  $2^{11/4}13^{1/2}\approx 24.255161140409015$
Ramified primes:   \(2\), \(13\) 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:  \(\Q(\sqrt{-26}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{4982081}a^{11}+\frac{1246874}{4982081}a^{10}+\frac{1521267}{4982081}a^{9}+\frac{570216}{4982081}a^{8}-\frac{292635}{4982081}a^{7}-\frac{1324712}{4982081}a^{6}+\frac{220280}{4982081}a^{5}-\frac{720810}{4982081}a^{4}-\frac{1799718}{4982081}a^{3}+\frac{2360419}{4982081}a^{2}+\frac{657788}{4982081}a-\frac{1311994}{4982081}$ 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:  $C_{2}$, 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:  $6$
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:   $a^{11}+8a^{9}-12a^{8}+10a^{7}-10a^{5}-16a^{3}+12a^{2}+8a$, $\frac{609502}{4982081}a^{11}+\frac{578927}{4982081}a^{10}+\frac{5166205}{4982081}a^{9}-\frac{2178128}{4982081}a^{8}+\frac{1864111}{4982081}a^{7}+\frac{6343841}{4982081}a^{6}-\frac{5982390}{4982081}a^{5}-\frac{5269878}{4982081}a^{4}-\frac{7018342}{4982081}a^{3}-\frac{5393194}{4982081}a^{2}+\frac{10061425}{4982081}a+\frac{890160}{4982081}$, $\frac{3285354}{4982081}a^{11}+\frac{1040685}{4982081}a^{10}+\frac{26426748}{4982081}a^{9}-\frac{30573642}{4982081}a^{8}+\frac{22459428}{4982081}a^{7}+\frac{13756555}{4982081}a^{6}-\frac{29199426}{4982081}a^{5}-\frac{8365415}{4982081}a^{4}-\frac{46748506}{4982081}a^{3}+\frac{23571910}{4982081}a^{2}+\frac{29018230}{4982081}a+\frac{2211218}{4982081}$, $\frac{4800920}{4982081}a^{11}-\frac{2370255}{4982081}a^{10}+\frac{39380338}{4982081}a^{9}-\frac{77164537}{4982081}a^{8}+\frac{84420691}{4982081}a^{7}-\frac{40547786}{4982081}a^{6}-\frac{29568756}{4982081}a^{5}+\frac{12281562}{4982081}a^{4}-\frac{78345500}{4982081}a^{3}+\frac{95787391}{4982081}a^{2}-\frac{9100591}{4982081}a+\frac{6988848}{4982081}$, $\frac{5672165}{4982081}a^{11}-\frac{2430013}{4982081}a^{10}+\frac{47657242}{4982081}a^{9}-\frac{87460856}{4982081}a^{8}+\frac{105563615}{4982081}a^{7}-\frac{50333928}{4982081}a^{6}-\frac{21480276}{4982081}a^{5}+\frac{16429405}{4982081}a^{4}-\frac{98175847}{4982081}a^{3}+\frac{110590109}{4982081}a^{2}-\frac{23300285}{4982081}a+\frac{7512634}{4982081}$, $\frac{6075132}{4982081}a^{11}-\frac{1205786}{4982081}a^{10}+\frac{47899786}{4982081}a^{9}-\frac{82805704}{4982081}a^{8}+\frac{69316192}{4982081}a^{7}-\frac{5682796}{4982081}a^{6}-\frac{66485422}{4982081}a^{5}+\frac{14090516}{4982081}a^{4}-\frac{88590064}{4982081}a^{3}+\frac{92684600}{4982081}a^{2}+\frac{41686240}{4982081}a-\frac{19212411}{4982081}$ 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:  \( 2211.66323272 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 2211.66323272 \cdot 1}{2\cdot\sqrt{3189382584467456}}\cr\approx \mathstrut & 0.766999403868 \end{aligned}\]

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 + 8*x^10 - 12*x^9 + 10*x^8 - 10*x^6 - 16*x^4 + 12*x^3 + 8*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^12 + 8*x^10 - 12*x^9 + 10*x^8 - 10*x^6 - 16*x^4 + 12*x^3 + 8*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^12 + 8*x^10 - 12*x^9 + 10*x^8 - 10*x^6 - 16*x^4 + 12*x^3 + 8*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^12 + 8*x^10 - 12*x^9 + 10*x^8 - 10*x^6 - 16*x^4 + 12*x^3 + 8*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

$S_4$ (as 12T8):

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); 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

3.1.104.1, 4.2.26624.2 x2, 6.2.2768896.2

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(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: 4.2.26624.2
Degree 6 siblings: 6.2.2768896.2, 6.0.4499456.1
Degree 8 sibling: 8.0.119793516544.13
Degree 12 sibling: 12.0.1295686674939904.1
Minimal sibling: 4.2.26624.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{4}$ ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{3}$ R ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(2\) Copy content Toggle raw display 2.1.4.11a1.4$x^{4} + 8 x^{3} + 8 x + 2$$4$$1$$11$$D_{4}$$$[3, 4]^{2}$$
2.2.4.22a1.13$x^{8} + 4 x^{7} + 18 x^{6} + 40 x^{5} + 67 x^{4} + 72 x^{3} + 66 x^{2} + 36 x + 19$$4$$2$$22$$D_4$$$[3, 4]^{2}$$
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.2$x^{2} + 26$$2$$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)