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Properties

Label 12.0.13196188560...1184.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{16}\cdot 61^{9}$
Root discriminant $267.12$
Ramified primes $2, 3, 61$
Class number $1122498$ (GRH)
Class group $[3, 3, 3, 41574]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14516726199, 154582308, 1955645292, 2865640, 111289299, 13800, 3301706, 36, 49098, -4, 354, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 354*x^10 - 4*x^9 + 49098*x^8 + 36*x^7 + 3301706*x^6 + 13800*x^5 + 111289299*x^4 + 2865640*x^3 + 1955645292*x^2 + 154582308*x + 14516726199)
 
gp: K = bnfinit(x^12 + 354*x^10 - 4*x^9 + 49098*x^8 + 36*x^7 + 3301706*x^6 + 13800*x^5 + 111289299*x^4 + 2865640*x^3 + 1955645292*x^2 + 154582308*x + 14516726199, 1)
 

Normalized defining polynomial

\( x^{12} + 354 x^{10} - 4 x^{9} + 49098 x^{8} + 36 x^{7} + 3301706 x^{6} + 13800 x^{5} + 111289299 x^{4} + 2865640 x^{3} + 1955645292 x^{2} + 154582308 x + 14516726199 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(131961885606929070005069021184=2^{18}\cdot 3^{16}\cdot 61^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $267.12$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4392=2^{3}\cdot 3^{2}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4392}(1,·)$, $\chi_{4392}(2917,·)$, $\chi_{4392}(2929,·)$, $\chi_{4392}(3049,·)$, $\chi_{4392}(1453,·)$, $\chi_{4392}(4381,·)$, $\chi_{4392}(1585,·)$, $\chi_{4392}(3061,·)$, $\chi_{4392}(121,·)$, $\chi_{4392}(1465,·)$, $\chi_{4392}(1597,·)$, $\chi_{4392}(133,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{10} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{3}{10} a^{2} + \frac{1}{5} a - \frac{3}{10}$, $\frac{1}{10} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{3}{10} a^{3} + \frac{1}{5} a^{2} - \frac{3}{10} a$, $\frac{1}{10} a^{8} - \frac{1}{5} a^{5} + \frac{3}{10} a^{4} - \frac{2}{5} a^{3} + \frac{3}{10} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{90} a^{9} - \frac{1}{30} a^{7} - \frac{1}{30} a^{6} - \frac{11}{30} a^{5} - \frac{1}{3} a^{4} + \frac{22}{45} a^{3} + \frac{11}{30} a^{2} - \frac{3}{10} a + \frac{1}{30}$, $\frac{1}{38605301868330} a^{10} + \frac{31903949683}{38605301868330} a^{9} + \frac{295671813667}{6434216978055} a^{8} + \frac{89758591643}{6434216978055} a^{7} - \frac{172222944751}{4289477985370} a^{6} - \frac{753920405621}{4289477985370} a^{5} - \frac{1620114187067}{7721060373666} a^{4} + \frac{1973044540679}{38605301868330} a^{3} - \frac{2359443922697}{6434216978055} a^{2} + \frac{1774283617688}{6434216978055} a + \frac{12560757643}{67728599769}$, $\frac{1}{78825466804026306232160910} a^{11} + \frac{55533483971}{15765093360805261246432182} a^{10} + \frac{43106523139069202501798}{39412733402013153116080455} a^{9} - \frac{193280230445491061837776}{13137577800671051038693485} a^{8} - \frac{202850919930041066360033}{5255031120268420415477394} a^{7} - \frac{236523900827558285402029}{26275155601342102077386970} a^{6} - \frac{212528366468572374529541}{736686605645105665721130} a^{5} + \frac{702203140626871893233849}{78825466804026306232160910} a^{4} + \frac{580212486665519511526507}{2074354389579639637688445} a^{3} - \frac{102017499803951746679033}{4379192600223683679564495} a^{2} - \frac{1360524982325285374964071}{13137577800671051038693485} a + \frac{328469924871094867471991}{691451463193213212562815}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{41574}$, which has order $1122498$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18105.42413749689 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\zeta_{9})^+\), 4.0.14526784.2, 6.6.1489222341.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.18.28$x^{12} - 52 x^{10} + 1100 x^{8} - 12000 x^{6} - 61072 x^{4} + 62144 x^{2} - 62144$$2$$6$$18$$C_{12}$$[3]^{6}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$61$61.12.9.1$x^{12} - 122 x^{8} - 1484679 x^{4} - 2269810000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$