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Properties

Label 12.12.184...000.1
Degree $12$
Signature $(12, 0)$
Discriminant $1.844\times 10^{18}$
Root discriminant \(33.28\)
Ramified primes $2,5,103$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_5$ (as 12T74)

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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 - 24*x^10 + 192*x^8 - 664*x^6 + 992*x^4 - 512*x^2 + 16)
 
Copy content gp:K = bnfinit(y^12 - 24*y^10 + 192*y^8 - 664*y^6 + 992*y^4 - 512*y^2 + 16, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 24*x^10 + 192*x^8 - 664*x^6 + 992*x^4 - 512*x^2 + 16);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 24*x^10 + 192*x^8 - 664*x^6 + 992*x^4 - 512*x^2 + 16)
 

\( x^{12} - 24x^{10} + 192x^{8} - 664x^{6} + 992x^{4} - 512x^{2} + 16 \) 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:  $(12, 0)$
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:   \(1844033634304000000\) \(\medspace = 2^{20}\cdot 5^{6}\cdot 103^{4}\) 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:  \(33.28\)
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:  $2^{13/6}5^{1/2}103^{1/2}\approx 101.89087030315206$
Ramified primes:   \(2\), \(5\), \(103\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{16}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a$, $\frac{1}{16}a^{8}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{9}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{32}a^{10}-\frac{1}{8}a^{4}-\frac{1}{2}a$, $\frac{1}{64}a^{11}-\frac{1}{32}a^{8}+\frac{1}{16}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{2}-\frac{1}{2}a$ 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 $1$ (assuming GRH)
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:  $C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
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:  $11$
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:   $\frac{1}{32}a^{11}-\frac{23}{32}a^{9}+\frac{85}{16}a^{7}-\frac{129}{8}a^{5}+\frac{155}{8}a^{3}-\frac{27}{4}a+\frac{1}{2}$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{3}{4}a^{9}-\frac{11}{16}a^{8}+\frac{97}{16}a^{7}+\frac{75}{16}a^{6}-\frac{175}{8}a^{5}-\frac{101}{8}a^{4}+\frac{143}{4}a^{3}+\frac{51}{4}a^{2}-\frac{81}{4}a-\frac{15}{4}$, $\frac{3}{64}a^{11}-\frac{1}{32}a^{10}-a^{9}+\frac{23}{32}a^{8}+\frac{101}{16}a^{7}-\frac{83}{16}a^{6}-\frac{221}{16}a^{5}+\frac{111}{8}a^{4}+\frac{25}{4}a^{3}-\frac{81}{8}a^{2}+\frac{5}{4}a+\frac{3}{4}$, $\frac{1}{64}a^{11}-\frac{1}{32}a^{10}-\frac{1}{4}a^{9}+\frac{19}{32}a^{8}+\frac{1}{2}a^{7}-\frac{23}{8}a^{6}+\frac{57}{16}a^{5}+\frac{27}{8}a^{4}-\frac{21}{2}a^{3}+\frac{7}{8}a^{2}+5a+\frac{1}{2}$, $\frac{3}{64}a^{11}-\frac{35}{32}a^{9}+\frac{1}{32}a^{8}+\frac{67}{8}a^{7}-\frac{5}{8}a^{6}-\frac{441}{16}a^{5}+\frac{7}{2}a^{4}+\frac{311}{8}a^{3}-\frac{51}{8}a^{2}-\frac{35}{2}a+3$, $\frac{1}{64}a^{11}+\frac{1}{32}a^{10}-\frac{1}{4}a^{9}-\frac{19}{32}a^{8}+\frac{1}{2}a^{7}+\frac{23}{8}a^{6}+\frac{57}{16}a^{5}-\frac{27}{8}a^{4}-\frac{21}{2}a^{3}-\frac{7}{8}a^{2}+5a-\frac{1}{2}$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{23}{32}a^{9}-\frac{11}{16}a^{8}+\frac{21}{4}a^{7}+\frac{75}{16}a^{6}-\frac{119}{8}a^{5}-\frac{101}{8}a^{4}+\frac{101}{8}a^{3}+\frac{49}{4}a^{2}+\frac{7}{2}a-\frac{5}{4}$, $\frac{1}{32}a^{10}+\frac{1}{16}a^{9}-\frac{5}{8}a^{8}-\frac{19}{16}a^{7}+\frac{57}{16}a^{6}+6a^{5}-\frac{61}{8}a^{4}-\frac{21}{2}a^{3}+5a^{2}+\frac{23}{4}a+\frac{3}{4}$, $\frac{1}{32}a^{10}-\frac{1}{16}a^{9}-\frac{5}{8}a^{8}+\frac{19}{16}a^{7}+\frac{57}{16}a^{6}-6a^{5}-\frac{61}{8}a^{4}+\frac{21}{2}a^{3}+5a^{2}-\frac{23}{4}a+\frac{3}{4}$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{3}{4}a^{9}-\frac{5}{8}a^{8}+\frac{95}{16}a^{7}+\frac{55}{16}a^{6}-\frac{157}{8}a^{5}-\frac{45}{8}a^{4}+\frac{105}{4}a^{3}-\frac{1}{2}a^{2}-\frac{47}{4}a+\frac{13}{4}$, $\frac{1}{64}a^{11}-\frac{1}{32}a^{10}-\frac{11}{32}a^{9}+\frac{23}{32}a^{8}+\frac{37}{16}a^{7}-\frac{21}{4}a^{6}-\frac{91}{16}a^{5}+\frac{119}{8}a^{4}+\frac{23}{8}a^{3}-\frac{101}{8}a^{2}+\frac{17}{4}a-\frac{5}{2}$ Copy content Toggle raw display (assuming GRH)
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:  \( 160256.354359 \) (assuming GRH)
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:  \( 10 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 160256.354359 \cdot 1}{2\cdot\sqrt{1844033634304000000}}\cr\approx \mathstrut & 0.241691174450 \end{aligned}\] (assuming GRH)

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 - 24*x^10 + 192*x^8 - 664*x^6 + 992*x^4 - 512*x^2 + 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 - 24*x^10 + 192*x^8 - 664*x^6 + 992*x^4 - 512*x^2 + 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 - 24*x^10 + 192*x^8 - 664*x^6 + 992*x^4 - 512*x^2 + 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 - 24*x^10 + 192*x^8 - 664*x^6 + 992*x^4 - 512*x^2 + 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

$S_5$ (as 12T74):

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 non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 6.6.1357952000.1

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 5 sibling: 5.5.13579520.1
Degree 6 sibling: 6.6.1357952000.1
Degree 10 siblings: 10.10.3688067268608000.1, 10.10.23050420428800000.1
Degree 15 sibling: deg 15
Degree 20 siblings: 20.20.8501150111111046013911040000000000.1, deg 20, deg 20
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 5.5.13579520.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.6.20a1.45$x^{12} + 6 x^{11} + 23 x^{10} + 60 x^{9} + 120 x^{8} + 186 x^{7} + 231 x^{6} + 228 x^{5} + 180 x^{4} + 110 x^{3} + 55 x^{2} + 20 x + 9$$6$$2$$20$$S_4$$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
\(5\) Copy content Toggle raw display 5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(103\) Copy content Toggle raw display 103.2.1.0a1.1$x^{2} + 102 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
103.2.1.0a1.1$x^{2} + 102 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
103.2.2.2a1.2$x^{4} + 204 x^{3} + 10414 x^{2} + 1020 x + 128$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
103.2.2.2a1.2$x^{4} + 204 x^{3} + 10414 x^{2} + 1020 x + 128$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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