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Properties

Label 12.0.56800822275...9472.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{8}\cdot 53^{9}$
Root discriminant $249.00$
Ramified primes $2, 3, 7, 53$
Class number $4516668$ (GRH)
Class group $[3, 6, 250926]$ (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![370936189, -117500690, 150014294, -28351612, 27197377, -8624434, 4485909, -202300, 77422, -1566, 475, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 475*x^10 - 1566*x^9 + 77422*x^8 - 202300*x^7 + 4485909*x^6 - 8624434*x^5 + 27197377*x^4 - 28351612*x^3 + 150014294*x^2 - 117500690*x + 370936189)
 
gp: K = bnfinit(x^12 - 4*x^11 + 475*x^10 - 1566*x^9 + 77422*x^8 - 202300*x^7 + 4485909*x^6 - 8624434*x^5 + 27197377*x^4 - 28351612*x^3 + 150014294*x^2 - 117500690*x + 370936189, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 475 x^{10} - 1566 x^{9} + 77422 x^{8} - 202300 x^{7} + 4485909 x^{6} - 8624434 x^{5} + 27197377 x^{4} - 28351612 x^{3} + 150014294 x^{2} - 117500690 x + 370936189 \)

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:  \(56800822275313340415461609472=2^{12}\cdot 3^{6}\cdot 7^{8}\cdot 53^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $249.00$
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:  $2, 3, 7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4452=2^{2}\cdot 3\cdot 7\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{4452}(1,·)$, $\chi_{4452}(3203,·)$, $\chi_{4452}(2437,·)$, $\chi_{4452}(529,·)$, $\chi_{4452}(1801,·)$, $\chi_{4452}(1355,·)$, $\chi_{4452}(2545,·)$, $\chi_{4452}(2627,·)$, $\chi_{4452}(659,·)$, $\chi_{4452}(23,·)$, $\chi_{4452}(3817,·)$, $\chi_{4452}(3263,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{21} a^{6} - \frac{2}{21} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{4}{21} a^{2} - \frac{4}{21} a - \frac{8}{21}$, $\frac{1}{21} a^{7} - \frac{10}{21} a^{5} - \frac{3}{7} a^{4} + \frac{2}{21} a^{3} + \frac{3}{7} a^{2} + \frac{5}{21} a + \frac{5}{21}$, $\frac{1}{147} a^{8} + \frac{2}{147} a^{7} + \frac{1}{49} a^{6} + \frac{50}{147} a^{5} - \frac{10}{147} a^{4} + \frac{31}{147} a^{3} + \frac{34}{147} a^{2} + \frac{26}{147} a + \frac{32}{147}$, $\frac{1}{1911} a^{9} - \frac{5}{1911} a^{8} - \frac{32}{1911} a^{7} + \frac{8}{1911} a^{6} - \frac{183}{637} a^{5} - \frac{319}{1911} a^{4} - \frac{47}{637} a^{3} + \frac{565}{1911} a^{2} - \frac{57}{637} a - \frac{5}{21}$, $\frac{1}{46127237415987} a^{10} - \frac{2664721970}{15375745805329} a^{9} + \frac{130731406322}{46127237415987} a^{8} + \frac{313504763269}{46127237415987} a^{7} + \frac{55064626075}{3548249031999} a^{6} + \frac{1039172509141}{3548249031999} a^{5} - \frac{17788245427588}{46127237415987} a^{4} - \frac{17504920172}{941372192163} a^{3} + \frac{3503711293573}{46127237415987} a^{2} - \frac{1694074947809}{15375745805329} a + \frac{928209367201}{3548249031999}$, $\frac{1}{4029622490401410433912563} a^{11} + \frac{1106824055}{309970960800108494916351} a^{10} - \frac{805290899196193662284}{4029622490401410433912563} a^{9} - \frac{2699580563835204080828}{4029622490401410433912563} a^{8} + \frac{10358655120243250359198}{1343207496800470144637521} a^{7} + \frac{93145234063429885435744}{4029622490401410433912563} a^{6} + \frac{481506839752949360453340}{1343207496800470144637521} a^{5} - \frac{91928609501645823800389}{191886785257210020662503} a^{4} - \frac{146096287747755817094821}{309970960800108494916351} a^{3} - \frac{76933612050054760560410}{575660355771630061987509} a^{2} - \frac{383430821618025874533082}{1343207496800470144637521} a + \frac{23103323500239463311590}{103323653600036164972117}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{250926}$, which has order $4516668$ (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:  \( 2414.7845504215825 \) (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

\(\Q(\sqrt{53}) \), \(\Q(\zeta_{7})^+\), 4.0.21438288.1, 6.6.357453677.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$3$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$53$53.12.9.1$x^{12} - 106 x^{8} - 716295 x^{4} - 609800192$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$