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Properties

Label 12.0.986...125.1
Degree $12$
Signature $[0, 6]$
Discriminant $9.861\times 10^{19}$
Root discriminant \(46.36\)
Ramified primes $3,5,19$
Class number $32$
Class group [2, 4, 4]
Galois group $C_2^4:C_{12}$ (as 12T99)

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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 - 3*x^11 + 24*x^10 - 24*x^9 + 273*x^8 - 114*x^7 + 1520*x^6 - 336*x^5 + 5496*x^4 - 363*x^3 + 10686*x^2 - 2670*x + 7921)
 
Copy content gp:K = bnfinit(y^12 - 3*y^11 + 24*y^10 - 24*y^9 + 273*y^8 - 114*y^7 + 1520*y^6 - 336*y^5 + 5496*y^4 - 363*y^3 + 10686*y^2 - 2670*y + 7921, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 24*x^10 - 24*x^9 + 273*x^8 - 114*x^7 + 1520*x^6 - 336*x^5 + 5496*x^4 - 363*x^3 + 10686*x^2 - 2670*x + 7921);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 + 24*x^10 - 24*x^9 + 273*x^8 - 114*x^7 + 1520*x^6 - 336*x^5 + 5496*x^4 - 363*x^3 + 10686*x^2 - 2670*x + 7921)
 

\( x^{12} - 3 x^{11} + 24 x^{10} - 24 x^{9} + 273 x^{8} - 114 x^{7} + 1520 x^{6} - 336 x^{5} + 5496 x^{4} + \cdots + 7921 \) 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:   \(98611378021423828125\) \(\medspace = 3^{18}\cdot 5^{9}\cdot 19^{4}\) 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:  \(46.36\)
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:  $3^{3/2}5^{3/4}19^{1/2}\approx 75.73317874146682$
Ramified primes:   \(3\), \(5\), \(19\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{53\cdots 71}a^{11}+\frac{26\cdots 84}{53\cdots 71}a^{10}+\frac{62\cdots 05}{53\cdots 71}a^{9}+\frac{60\cdots 69}{53\cdots 71}a^{8}+\frac{48\cdots 61}{53\cdots 71}a^{7}-\frac{20\cdots 13}{53\cdots 71}a^{6}+\frac{14\cdots 53}{53\cdots 71}a^{5}-\frac{37\cdots 41}{53\cdots 71}a^{4}+\frac{23\cdots 25}{53\cdots 71}a^{3}-\frac{19\cdots 88}{53\cdots 71}a^{2}+\frac{16\cdots 80}{53\cdots 71}a+\frac{23\cdots 53}{60\cdots 39}$ 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:  $C_{2}\times C_{4}\times C_{4}$, which has order $32$
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:  $C_{2}\times C_{4}\times C_{4}$, which has order $32$
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:  $5$
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:   \( -1 \)  (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:   $\frac{1203248218130}{31\cdots 81}a^{11}+\frac{3032362189065}{31\cdots 81}a^{10}-\frac{21650889935335}{31\cdots 81}a^{9}-\frac{5912954843030}{31\cdots 81}a^{8}-\frac{180501260702530}{31\cdots 81}a^{7}-\frac{192363699674320}{31\cdots 81}a^{6}-\frac{523388658879894}{31\cdots 81}a^{5}-\frac{14\cdots 85}{31\cdots 81}a^{4}-\frac{980739170277990}{31\cdots 81}a^{3}-\frac{40\cdots 95}{31\cdots 81}a^{2}+\frac{830340098511580}{31\cdots 81}a-\frac{48\cdots 69}{31\cdots 81}$, $\frac{833153665010}{31\cdots 81}a^{11}+\frac{2298544316839}{31\cdots 81}a^{10}-\frac{16030058005430}{31\cdots 81}a^{9}+\frac{8257395488325}{31\cdots 81}a^{8}-\frac{172567874875670}{31\cdots 81}a^{7}+\frac{113940069257130}{31\cdots 81}a^{6}-\frac{861787887802932}{31\cdots 81}a^{5}+\frac{804075921124220}{31\cdots 81}a^{4}-\frac{29\cdots 95}{31\cdots 81}a^{3}+\frac{30\cdots 30}{31\cdots 81}a^{2}-\frac{30\cdots 50}{31\cdots 81}a+\frac{74\cdots 76}{31\cdots 81}$, $\frac{10\cdots 83}{53\cdots 71}a^{11}-\frac{17\cdots 21}{53\cdots 71}a^{10}+\frac{59\cdots 04}{53\cdots 71}a^{9}-\frac{48\cdots 74}{53\cdots 71}a^{8}+\frac{41\cdots 76}{53\cdots 71}a^{7}-\frac{43\cdots 13}{53\cdots 71}a^{6}+\frac{27\cdots 82}{53\cdots 71}a^{5}-\frac{19\cdots 06}{53\cdots 71}a^{4}+\frac{40\cdots 27}{53\cdots 71}a^{3}-\frac{54\cdots 61}{53\cdots 71}a^{2}+\frac{13\cdots 20}{53\cdots 71}a-\frac{65\cdots 50}{60\cdots 39}$, $\frac{22\cdots 39}{53\cdots 71}a^{11}-\frac{10\cdots 95}{53\cdots 71}a^{10}+\frac{42\cdots 63}{53\cdots 71}a^{9}+\frac{62\cdots 02}{53\cdots 71}a^{8}+\frac{59\cdots 28}{53\cdots 71}a^{7}+\frac{10\cdots 73}{53\cdots 71}a^{6}+\frac{38\cdots 66}{53\cdots 71}a^{5}+\frac{56\cdots 50}{53\cdots 71}a^{4}+\frac{12\cdots 12}{53\cdots 71}a^{3}+\frac{15\cdots 86}{53\cdots 71}a^{2}+\frac{19\cdots 51}{53\cdots 71}a+\frac{18\cdots 21}{60\cdots 39}$, $\frac{35\cdots 07}{53\cdots 71}a^{11}+\frac{15\cdots 72}{53\cdots 71}a^{10}-\frac{87\cdots 60}{53\cdots 71}a^{9}+\frac{12\cdots 61}{53\cdots 71}a^{8}-\frac{65\cdots 31}{53\cdots 71}a^{7}+\frac{63\cdots 68}{53\cdots 71}a^{6}-\frac{27\cdots 93}{53\cdots 71}a^{5}+\frac{13\cdots 10}{53\cdots 71}a^{4}-\frac{68\cdots 73}{53\cdots 71}a^{3}+\frac{37\cdots 32}{53\cdots 71}a^{2}-\frac{59\cdots 37}{53\cdots 71}a+\frac{11\cdots 94}{60\cdots 39}$ 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:  \( 3153.9783425305222 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3153.9783425305222 \cdot 32}{2\cdot\sqrt{98611378021423828125}}\cr\approx \mathstrut & 0.312675883550619 \end{aligned}\]

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 + 24*x^10 - 24*x^9 + 273*x^8 - 114*x^7 + 1520*x^6 - 336*x^5 + 5496*x^4 - 363*x^3 + 10686*x^2 - 2670*x + 7921) 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 + 24*x^10 - 24*x^9 + 273*x^8 - 114*x^7 + 1520*x^6 - 336*x^5 + 5496*x^4 - 363*x^3 + 10686*x^2 - 2670*x + 7921, 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 + 24*x^10 - 24*x^9 + 273*x^8 - 114*x^7 + 1520*x^6 - 336*x^5 + 5496*x^4 - 363*x^3 + 10686*x^2 - 2670*x + 7921); 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 + 24*x^10 - 24*x^9 + 273*x^8 - 114*x^7 + 1520*x^6 - 336*x^5 + 5496*x^4 - 363*x^3 + 10686*x^2 - 2670*x + 7921); 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

$C_2^4:C_{12}$ (as 12T99):

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 192
The 20 conjugacy class representatives for $C_2^4:C_{12}$
Character table for $C_2^4:C_{12}$

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.2.296065125.2

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(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
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 ${\href{/padicField/2.12.0.1}{12} }$ R R ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ R ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(3\) Copy content Toggle raw display 3.2.6.18a4.3$x^{12} + 12 x^{11} + 72 x^{10} + 283 x^{9} + 810 x^{8} + 1776 x^{7} + 3056 x^{6} + 4152 x^{5} + 4416 x^{4} + 3590 x^{3} + 2130 x^{2} + 843 x + 190$$6$$2$$18$$C_{12}$$$[2]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.3.4.9a1.1$x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 491 x^{2} + 324 x + 81$$4$$3$$9$$C_{12}$$$[\ ]_{4}^{3}$$
\(19\) Copy content Toggle raw display $\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.2.2.2a1.2$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$$2$$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)