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Properties

Label 12.0.30484209987...2169.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{10}\cdot 23^{6}$
Root discriminant $42.04$
Ramified primes $3, 7, 23$
Class number $234$ (GRH)
Class group $[234]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![213487, -37447, 110223, -25046, 27208, -6772, 4353, -976, 461, -84, 29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 29*x^10 - 84*x^9 + 461*x^8 - 976*x^7 + 4353*x^6 - 6772*x^5 + 27208*x^4 - 25046*x^3 + 110223*x^2 - 37447*x + 213487)
 
gp: K = bnfinit(x^12 - 4*x^11 + 29*x^10 - 84*x^9 + 461*x^8 - 976*x^7 + 4353*x^6 - 6772*x^5 + 27208*x^4 - 25046*x^3 + 110223*x^2 - 37447*x + 213487, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 29 x^{10} - 84 x^{9} + 461 x^{8} - 976 x^{7} + 4353 x^{6} - 6772 x^{5} + 27208 x^{4} - 25046 x^{3} + 110223 x^{2} - 37447 x + 213487 \)

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:  \(30484209987928082169=3^{6}\cdot 7^{10}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.04$
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:  $3, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(483=3\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{483}(1,·)$, $\chi_{483}(482,·)$, $\chi_{483}(68,·)$, $\chi_{483}(298,·)$, $\chi_{483}(461,·)$, $\chi_{483}(206,·)$, $\chi_{483}(47,·)$, $\chi_{483}(436,·)$, $\chi_{483}(277,·)$, $\chi_{483}(22,·)$, $\chi_{483}(185,·)$, $\chi_{483}(415,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41} a^{10} - \frac{12}{41} a^{9} - \frac{13}{41} a^{8} - \frac{5}{41} a^{7} - \frac{1}{41} a^{6} + \frac{9}{41} a^{5} - \frac{9}{41} a^{4} + \frac{12}{41} a^{3} - \frac{18}{41} a^{2} + \frac{10}{41} a$, $\frac{1}{108256666399667565163} a^{11} - \frac{796763704305912062}{108256666399667565163} a^{10} + \frac{5330350005907312499}{108256666399667565163} a^{9} + \frac{36436298911572044714}{108256666399667565163} a^{8} + \frac{38158096991979667281}{108256666399667565163} a^{7} + \frac{31150650665589635112}{108256666399667565163} a^{6} - \frac{51612432065875835360}{108256666399667565163} a^{5} - \frac{10305901420836746729}{108256666399667565163} a^{4} + \frac{24521177445680892148}{108256666399667565163} a^{3} - \frac{51927535761764859783}{108256666399667565163} a^{2} - \frac{52190521113915500995}{108256666399667565163} a - \frac{273457044334393182}{2640406497552867443}$

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

Class group and class number

$C_{234}$, which has order $234$ (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:  \( 140.798796005 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-483}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-23})\), \(\Q(\zeta_{21})^+\), 6.0.29212967.1, 6.0.5521250763.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$23$23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$