Properties

Label 12.4.601013506097152.2
Degree $12$
Signature $(4, 4)$
Discriminant $6.010\times 10^{14}$
Root discriminant \(17.04\)
Ramified primes $2,23,37$
Class number $1$
Class group trivial
Galois group $C_2^5:S_4$ (as 12T190)

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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 - 4*x^11 - 2*x^10 + 14*x^9 + 24*x^8 - 76*x^7 + 48*x^6 + 64*x^5 - 196*x^4 + 8*x^3 + 136*x^2 + 40*x + 4)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 - 2*y^10 + 14*y^9 + 24*y^8 - 76*y^7 + 48*y^6 + 64*y^5 - 196*y^4 + 8*y^3 + 136*y^2 + 40*y + 4, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 2*x^10 + 14*x^9 + 24*x^8 - 76*x^7 + 48*x^6 + 64*x^5 - 196*x^4 + 8*x^3 + 136*x^2 + 40*x + 4);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 2*x^10 + 14*x^9 + 24*x^8 - 76*x^7 + 48*x^6 + 64*x^5 - 196*x^4 + 8*x^3 + 136*x^2 + 40*x + 4)
 

\( x^{12} - 4 x^{11} - 2 x^{10} + 14 x^{9} + 24 x^{8} - 76 x^{7} + 48 x^{6} + 64 x^{5} - 196 x^{4} + \cdots + 4 \) 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:   \(601013506097152\) \(\medspace = 2^{14}\cdot 23^{2}\cdot 37^{5}\) 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:  \(17.04\)
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:  $2^{7/6}23^{1/2}37^{1/2}\approx 65.4887108885352$
Ramified primes:   \(2\), \(23\), \(37\) 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:  \(\Q(\sqrt{37}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{769356628}a^{11}+\frac{27232160}{192339157}a^{10}-\frac{23360459}{192339157}a^{9}-\frac{21202639}{192339157}a^{8}+\frac{30072858}{192339157}a^{7}+\frac{90806009}{384678314}a^{6}+\frac{185581111}{384678314}a^{5}-\frac{65232705}{192339157}a^{4}+\frac{12722667}{192339157}a^{3}-\frac{2271806}{192339157}a^{2}-\frac{33579888}{192339157}a-\frac{16211949}{192339157}$ 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:  $7$
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:   $\frac{4189050}{192339157}a^{11}-\frac{12405527}{192339157}a^{10}-\frac{27258588}{192339157}a^{9}+\frac{111932691}{384678314}a^{8}+\frac{158543498}{192339157}a^{7}-\frac{456416923}{384678314}a^{6}-\frac{130529039}{192339157}a^{5}+\frac{599056365}{192339157}a^{4}-\frac{745032789}{192339157}a^{3}-\frac{800796173}{192339157}a^{2}+\frac{920347869}{192339157}a+\frac{317911778}{192339157}$, $\frac{4849503}{192339157}a^{11}-\frac{30214303}{384678314}a^{10}-\frac{26255276}{192339157}a^{9}+\frac{127608663}{384678314}a^{8}+\frac{152848402}{192339157}a^{7}-\frac{282663912}{192339157}a^{6}-\frac{29084031}{192339157}a^{5}+\frac{671101236}{192339157}a^{4}-\frac{926026778}{192339157}a^{3}-\frac{541754460}{192339157}a^{2}+\frac{796958138}{192339157}a+\frac{133606851}{192339157}$, $\frac{18664815}{769356628}a^{11}-\frac{35147317}{384678314}a^{10}-\frac{9728703}{192339157}a^{9}+\frac{58916268}{192339157}a^{8}+\frac{207383927}{384678314}a^{7}-\frac{674834733}{384678314}a^{6}+\frac{456539837}{384678314}a^{5}+\frac{308877088}{192339157}a^{4}-\frac{890527834}{192339157}a^{3}+\frac{59507173}{192339157}a^{2}+\frac{711629975}{192339157}a+\frac{126075204}{192339157}$, $\frac{36067507}{769356628}a^{11}-\frac{72908293}{384678314}a^{10}-\frac{24818115}{384678314}a^{9}+\frac{116922566}{192339157}a^{8}+\frac{193851302}{192339157}a^{7}-\frac{1338041611}{384678314}a^{6}+\frac{1118454167}{384678314}a^{5}+\frac{384898099}{192339157}a^{4}-\frac{1724836407}{192339157}a^{3}+\frac{410144242}{192339157}a^{2}+\frac{669640596}{192339157}a+\frac{109319004}{192339157}$, $\frac{17904295}{769356628}a^{11}-\frac{20318991}{192339157}a^{10}+\frac{14253245}{384678314}a^{9}+\frac{96385435}{384678314}a^{8}+\frac{46715095}{192339157}a^{7}-\frac{701800249}{384678314}a^{6}+\frac{1168087031}{384678314}a^{5}-\frac{107116264}{192339157}a^{4}-\frac{919602632}{192339157}a^{3}+\frac{800115590}{192339157}a^{2}+\frac{94785275}{192339157}a-\frac{71791644}{192339157}$, $\frac{73895357}{769356628}a^{11}-\frac{137962087}{384678314}a^{10}-\frac{90040433}{384678314}a^{9}+\frac{421927395}{384678314}a^{8}+\frac{464896003}{192339157}a^{7}-\frac{1186360426}{192339157}a^{6}+\frac{1742459849}{384678314}a^{5}+\frac{888742936}{192339157}a^{4}-\frac{2970979956}{192339157}a^{3}-\frac{183444415}{192339157}a^{2}+\frac{1196298732}{192339157}a+\frac{142458533}{192339157}$, $\frac{15769157}{769356628}a^{11}-\frac{48574471}{384678314}a^{10}+\frac{71926821}{384678314}a^{9}+\frac{29547545}{192339157}a^{8}-\frac{33045271}{384678314}a^{7}-\frac{806492095}{384678314}a^{6}+\frac{2014705867}{384678314}a^{5}-\frac{911012896}{192339157}a^{4}-\frac{356197626}{192339157}a^{3}+\frac{1395251906}{192339157}a^{2}-\frac{670204228}{192339157}a-\frac{216834658}{192339157}$ 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:  \( 792.557510465 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 792.557510465 \cdot 1}{2\cdot\sqrt{601013506097152}}\cr\approx \mathstrut & 0.403086386029 \end{aligned}\]

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 - 4*x^11 - 2*x^10 + 14*x^9 + 24*x^8 - 76*x^7 + 48*x^6 + 64*x^5 - 196*x^4 + 8*x^3 + 136*x^2 + 40*x + 4) 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 - 4*x^11 - 2*x^10 + 14*x^9 + 24*x^8 - 76*x^7 + 48*x^6 + 64*x^5 - 196*x^4 + 8*x^3 + 136*x^2 + 40*x + 4, 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 - 4*x^11 - 2*x^10 + 14*x^9 + 24*x^8 - 76*x^7 + 48*x^6 + 64*x^5 - 196*x^4 + 8*x^3 + 136*x^2 + 40*x + 4); 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 - 4*x^11 - 2*x^10 + 14*x^9 + 24*x^8 - 76*x^7 + 48*x^6 + 64*x^5 - 196*x^4 + 8*x^3 + 136*x^2 + 40*x + 4); 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

$C_2^5:S_4$ (as 12T190):

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 768
The 23 conjugacy class representatives for $C_2^5:S_4$
Character table for $C_2^5:S_4$

Intermediate fields

3.3.148.1, 6.2.87616.1

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 12 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 12.0.4060902068224.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{3}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ R ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ R ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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 2.1.12.14a1.1$x^{12} + 2 x^{3} + 2$$12$$1$$14$$S_4$$$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$$
\(23\) Copy content Toggle raw display 23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.4.1.0a1.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
23.2.2.2a1.2$x^{4} + 42 x^{3} + 451 x^{2} + 210 x + 48$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(37\) Copy content Toggle raw display 37.1.2.1a1.2$x^{2} + 74$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.4.2.4a1.2$x^{8} + 12 x^{6} + 48 x^{5} + 40 x^{4} + 288 x^{3} + 600 x^{2} + 96 x + 41$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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