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Properties

Label 12.4.1240804736328125.1
Degree $12$
Signature $[4, 4]$
Discriminant $1.241\times 10^{15}$
Root discriminant \(18.11\)
Ramified primes $5,71$
Class number $2$
Class group [2]
Galois group $S_4^2:C_4$ (as 12T237)

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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^11 + 11*x^10 - 20*x^9 + 30*x^8 - 52*x^7 - 11*x^6 + 3*x^5 - 65*x^4 + 80*x^3 - 59*x^2 - 12*x + 1)
 
Copy content gp:K = bnfinit(y^12 - 2*y^11 + 11*y^10 - 20*y^9 + 30*y^8 - 52*y^7 - 11*y^6 + 3*y^5 - 65*y^4 + 80*y^3 - 59*y^2 - 12*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 11*x^10 - 20*x^9 + 30*x^8 - 52*x^7 - 11*x^6 + 3*x^5 - 65*x^4 + 80*x^3 - 59*x^2 - 12*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 + 11*x^10 - 20*x^9 + 30*x^8 - 52*x^7 - 11*x^6 + 3*x^5 - 65*x^4 + 80*x^3 - 59*x^2 - 12*x + 1)
 

\( x^{12} - 2 x^{11} + 11 x^{10} - 20 x^{9} + 30 x^{8} - 52 x^{7} - 11 x^{6} + 3 x^{5} - 65 x^{4} + \cdots + 1 \) 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:  $[4, 4]$
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:   \(1240804736328125\) \(\medspace = 5^{11}\cdot 71^{4}\) 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:  \(18.11\)
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:  $5^{11/12}71^{3/4}\approx 106.94642350747239$
Ramified primes:   \(5\), \(71\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{1643293409}a^{11}-\frac{558653895}{1643293409}a^{10}+\frac{788035058}{1643293409}a^{9}+\frac{158917792}{1643293409}a^{8}+\frac{692978894}{1643293409}a^{7}-\frac{262585762}{1643293409}a^{6}+\frac{305038149}{1643293409}a^{5}-\frac{328767758}{1643293409}a^{4}+\frac{223547089}{1643293409}a^{3}-\frac{170223947}{1643293409}a^{2}+\frac{177357141}{1643293409}a-\frac{258156454}{1643293409}$ 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:  $C_{2}$, which has order $2$
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 $4$
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:   $\frac{25656}{684991}a^{11}-\frac{37074}{684991}a^{10}+\frac{247224}{684991}a^{9}-\frac{338756}{684991}a^{8}+\frac{431102}{684991}a^{7}-\frac{810043}{684991}a^{6}-\frac{1098761}{684991}a^{5}-\frac{133754}{684991}a^{4}-\frac{1408984}{684991}a^{3}+\frac{709873}{684991}a^{2}+\frac{154912}{684991}a-\frac{435922}{684991}$, $\frac{229531150}{1643293409}a^{11}+\frac{445927331}{1643293409}a^{10}-\frac{2518980862}{1643293409}a^{9}+\frac{4496530814}{1643293409}a^{8}-\frac{6850967970}{1643293409}a^{7}+\frac{12066409011}{1643293409}a^{6}+\frac{2417793016}{1643293409}a^{5}+\frac{810537515}{1643293409}a^{4}+\frac{14351519058}{1643293409}a^{3}-\frac{18127837180}{1643293409}a^{2}+\frac{13434140776}{1643293409}a+\frac{954224023}{1643293409}$, $\frac{178772022}{1643293409}a^{11}+\frac{334872992}{1643293409}a^{10}-\frac{1946166543}{1643293409}a^{9}+\frac{3374360168}{1643293409}a^{8}-\frac{5184636021}{1643293409}a^{7}+\frac{9138999755}{1643293409}a^{6}+\frac{2486912517}{1643293409}a^{5}+\frac{1110552246}{1643293409}a^{4}+\frac{11993120411}{1643293409}a^{3}-\frac{13644100668}{1643293409}a^{2}+\frac{8604427081}{1643293409}a+\frac{294079265}{1643293409}$, $\frac{399433444}{1643293409}a^{11}+\frac{753424731}{1643293409}a^{10}-\frac{4336973777}{1643293409}a^{9}+\frac{7532614082}{1643293409}a^{8}-\frac{11320771534}{1643293409}a^{7}+\frac{19674479456}{1643293409}a^{6}+\frac{6596124137}{1643293409}a^{5}-\frac{284587066}{1643293409}a^{4}+\frac{27105375043}{1643293409}a^{3}-\frac{28686365745}{1643293409}a^{2}+\frac{21647300486}{1643293409}a+\frac{6832206875}{1643293409}$, $\frac{13134969}{1643293409}a^{11}-\frac{5861788}{1643293409}a^{10}+\frac{94092186}{1643293409}a^{9}-\frac{34966530}{1643293409}a^{8}-\frac{130789211}{1643293409}a^{7}+\frac{107049634}{1643293409}a^{6}-\frac{1499130965}{1643293409}a^{5}+\frac{568005692}{1643293409}a^{4}-\frac{234654820}{1643293409}a^{3}+\frac{108197074}{1643293409}a^{2}+\frac{1800149777}{1643293409}a-\frac{1872824559}{1643293409}$, $\frac{46625576}{1643293409}a^{11}-\frac{102719277}{1643293409}a^{10}+\frac{533576964}{1643293409}a^{9}-\frac{1041933943}{1643293409}a^{8}+\frac{1553350177}{1643293409}a^{7}-\frac{2755301222}{1643293409}a^{6}-\frac{165497092}{1643293409}a^{5}-\frac{76210718}{1643293409}a^{4}-\frac{2271827167}{1643293409}a^{3}+\frac{3718560409}{1643293409}a^{2}-\frac{2235931584}{1643293409}a+\frac{312184338}{1643293409}$, $\frac{132146446}{1643293409}a^{11}+\frac{232153715}{1643293409}a^{10}-\frac{1412589579}{1643293409}a^{9}+\frac{2332426225}{1643293409}a^{8}-\frac{3631285844}{1643293409}a^{7}+\frac{6383698533}{1643293409}a^{6}+\frac{2321415425}{1643293409}a^{5}+\frac{1034341528}{1643293409}a^{4}+\frac{9721293244}{1643293409}a^{3}-\frac{9925540259}{1643293409}a^{2}+\frac{6368495497}{1643293409}a+\frac{2249557012}{1643293409}$ 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:  \( 418.7100827562533 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 418.7100827562533 \cdot 2}{2\cdot\sqrt{1240804736328125}}\cr\approx \mathstrut & 0.296415666577212 \end{aligned}\]

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^11 + 11*x^10 - 20*x^9 + 30*x^8 - 52*x^7 - 11*x^6 + 3*x^5 - 65*x^4 + 80*x^3 - 59*x^2 - 12*x + 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 - 2*x^11 + 11*x^10 - 20*x^9 + 30*x^8 - 52*x^7 - 11*x^6 + 3*x^5 - 65*x^4 + 80*x^3 - 59*x^2 - 12*x + 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 - 2*x^11 + 11*x^10 - 20*x^9 + 30*x^8 - 52*x^7 - 11*x^6 + 3*x^5 - 65*x^4 + 80*x^3 - 59*x^2 - 12*x + 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 = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 + 11*x^10 - 20*x^9 + 30*x^8 - 52*x^7 - 11*x^6 + 3*x^5 - 65*x^4 + 80*x^3 - 59*x^2 - 12*x + 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

$S_4^2:C_4$ (as 12T237):

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 solvable group of order 2304
The 40 conjugacy class representatives for $S_4^2:C_4$
Character table for $S_4^2:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 6.4.221875.1

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(b)]
 
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 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 siblings: 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 ${\href{/padicField/2.12.0.1}{12} }$ ${\href{/padicField/3.12.0.1}{12} }$ R ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.4.0.1}{4} }^{3}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(5\) Copy content Toggle raw display 5.1.12.11a1.1$x^{12} + 5$$12$$1$$11$$S_3 \times C_4$$$[\ ]_{12}^{2}$$
\(71\) Copy content Toggle raw display 71.2.1.0a1.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$$[\ ]^{2}$$
71.1.2.1a1.1$x^{2} + 71$$2$$1$$1$$C_2$$$[\ ]_{2}$$
71.1.4.3a1.2$x^{4} + 497$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
71.4.1.0a1.1$x^{4} + 4 x^{2} + 41 x + 7$$1$$4$$0$$C_4$$$[\ ]^{4}$$

Spectrum of ring of integers

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