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Properties

Label 12.0.144054149089536.2
Degree $12$
Signature $[0, 6]$
Discriminant $1.441\times 10^{14}$
Root discriminant \(15.13\)
Ramified primes $2,3,7$
Class number $1$
Class group trivial
Galois group $D_6$ (as 12T3)

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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 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*x + 4)
 
Copy content gp:K = bnfinit(y^12 - 3*y^11 + y^9 + 48*y^8 - 189*y^7 + 431*y^6 - 654*y^5 + 624*y^4 - 340*y^3 + 96*y^2 - 12*y + 4, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*x + 4);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*x + 4)
 

\( x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 \) 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:  $[0, 6]$
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:   \(144054149089536\) \(\medspace = 2^{8}\cdot 3^{14}\cdot 7^{6}\) 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 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:  $2^{2/3}3^{7/6}7^{1/2}\approx 15.13133155729692$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $D_6$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 6.0.1714608.1$^{9}$, 6.0.4000752.1$^{9}$, 12.0.144054149089536.2$^{12}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7}a^{8}+\frac{1}{7}a^{6}-\frac{1}{7}a^{4}+\frac{2}{7}a^{2}-\frac{3}{7}$, $\frac{1}{14}a^{9}-\frac{1}{14}a^{8}-\frac{3}{7}a^{7}-\frac{1}{14}a^{6}+\frac{3}{7}a^{5}+\frac{1}{14}a^{4}-\frac{5}{14}a^{3}-\frac{1}{7}a^{2}+\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{196}a^{10}-\frac{5}{196}a^{9}-\frac{3}{49}a^{8}+\frac{37}{196}a^{7}-\frac{2}{7}a^{6}+\frac{89}{196}a^{5}+\frac{15}{196}a^{4}-\frac{47}{98}a^{3}-\frac{2}{49}a^{2}+\frac{39}{98}a-\frac{20}{49}$, $\frac{1}{5500348}a^{11}-\frac{169}{1375087}a^{10}-\frac{134375}{5500348}a^{9}-\frac{6575}{289492}a^{8}-\frac{2359467}{5500348}a^{7}-\frac{1483029}{5500348}a^{6}+\frac{628990}{1375087}a^{5}-\frac{109535}{289492}a^{4}+\frac{49195}{1375087}a^{3}-\frac{327}{56126}a^{2}-\frac{610051}{2750174}a-\frac{96907}{1375087}$ 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:  Trivial group, which has order $1$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   $1$

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:  $5$
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:   \( \frac{559}{10339} a^{11} - \frac{4785}{41356} a^{10} - \frac{901}{5908} a^{9} + \frac{47}{1477} a^{8} + \frac{113135}{41356} a^{7} - \frac{80932}{10339} a^{6} + \frac{580379}{41356} a^{5} - \frac{648191}{41356} a^{4} + \frac{106179}{20678} a^{3} + \frac{52092}{10339} a^{2} - \frac{71077}{20678} a + \frac{6829}{10339} \)  (order $6$) 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:   $\frac{32017}{1375087}a^{11}-\frac{98027}{2750174}a^{10}-\frac{312635}{2750174}a^{9}-\frac{873}{72373}a^{8}+\frac{3482531}{2750174}a^{7}-\frac{3528439}{1375087}a^{6}+\frac{8721281}{2750174}a^{5}-\frac{236203}{144746}a^{4}-\frac{5703407}{1375087}a^{3}+\frac{885212}{196441}a^{2}+\frac{1514361}{1375087}a-\frac{188745}{1375087}$, $\frac{269483}{1375087}a^{11}-\frac{3169293}{5500348}a^{10}-\frac{246693}{5500348}a^{9}+\frac{24771}{144746}a^{8}+\frac{51993911}{5500348}a^{7}-\frac{99875645}{2750174}a^{6}+\frac{451499123}{5500348}a^{5}-\frac{35743979}{289492}a^{4}+\frac{158004362}{1375087}a^{3}-\frac{12182946}{196441}a^{2}+\frac{44641455}{2750174}a-\frac{1205566}{1375087}$, $\frac{202453}{2750174}a^{11}-\frac{320213}{1375087}a^{10}+\frac{35635}{1375087}a^{9}+\frac{8149}{72373}a^{8}+\frac{9682993}{2750174}a^{7}-\frac{19990102}{1375087}a^{6}+\frac{46076706}{1375087}a^{5}-\frac{3690054}{72373}a^{4}+\frac{134186735}{2750174}a^{3}-\frac{5053920}{196441}a^{2}+\frac{9103459}{1375087}a-\frac{539142}{1375087}$, $\frac{547915}{5500348}a^{11}-\frac{2793303}{5500348}a^{10}+\frac{472224}{1375087}a^{9}+\frac{213401}{289492}a^{8}+\frac{7024824}{1375087}a^{7}-\frac{158684065}{5500348}a^{6}+\frac{377684933}{5500348}a^{5}-\frac{16215729}{144746}a^{4}+\frac{153896039}{1375087}a^{3}-\frac{20335523}{392882}a^{2}+\frac{3773558}{1375087}a+\frac{473757}{1375087}$, $\frac{208685}{5500348}a^{11}-\frac{189015}{1375087}a^{10}+\frac{411009}{5500348}a^{9}+\frac{11401}{289492}a^{8}+\frac{9823137}{5500348}a^{7}-\frac{45757553}{5500348}a^{6}+\frac{28732919}{1375087}a^{5}-\frac{10214455}{289492}a^{4}+\frac{54525757}{1375087}a^{3}-\frac{10294709}{392882}a^{2}+\frac{22430477}{2750174}a-\frac{165983}{1375087}$ 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:  \( 812.119885731 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 812.119885731 \cdot 1}{6\cdot\sqrt{144054149089536}}\cr\approx \mathstrut & 0.693881356553 \end{aligned}\]

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 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*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 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*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 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*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 = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*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

$D_6$ (as 12T3):

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 12
The 6 conjugacy class representatives for $D_6$
Character table for $D_6$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 3.3.756.1 x3, \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.6.12002256.1, 6.0.1714608.1 x3, 6.0.4000752.1 x3

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 6 siblings: 6.0.1714608.1, 6.0.4000752.1
Minimal sibling: 6.0.1714608.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.2.6.14a2.1$x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3126 x^{4} + 2264 x^{3} + 1200 x^{2} + 432 x + 91$$6$$2$$14$$D_6$$$[\frac{3}{2}]_{2}^{2}$$
\(7\) Copy content Toggle raw display 7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$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)