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Properties

Label 12.0.11201589161...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 5^{9}\cdot 23^{6}$
Root discriminant $83.32$
Ramified primes $3, 5, 23$
Class number $17540$ (GRH)
Class group $[2, 8770]$ (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![76909321, -17829192, 21932874, -3206849, 3288795, -137187, 257648, -1701, 9774, -1, 168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 168*x^10 - x^9 + 9774*x^8 - 1701*x^7 + 257648*x^6 - 137187*x^5 + 3288795*x^4 - 3206849*x^3 + 21932874*x^2 - 17829192*x + 76909321)
 
gp: K = bnfinit(x^12 + 168*x^10 - x^9 + 9774*x^8 - 1701*x^7 + 257648*x^6 - 137187*x^5 + 3288795*x^4 - 3206849*x^3 + 21932874*x^2 - 17829192*x + 76909321, 1)
 

Normalized defining polynomial

\( x^{12} + 168 x^{10} - x^{9} + 9774 x^{8} - 1701 x^{7} + 257648 x^{6} - 137187 x^{5} + 3288795 x^{4} - 3206849 x^{3} + 21932874 x^{2} - 17829192 x + 76909321 \)

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:  \(112015891613143986328125=3^{18}\cdot 5^{9}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.32$
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:  $3, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1035=3^{2}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1035}(1,·)$, $\chi_{1035}(482,·)$, $\chi_{1035}(68,·)$, $\chi_{1035}(137,·)$, $\chi_{1035}(139,·)$, $\chi_{1035}(413,·)$, $\chi_{1035}(691,·)$, $\chi_{1035}(758,·)$, $\chi_{1035}(484,·)$, $\chi_{1035}(346,·)$, $\chi_{1035}(827,·)$, $\chi_{1035}(829,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{19} a^{8} + \frac{2}{19} a^{7} - \frac{8}{19} a^{6} + \frac{9}{19} a^{5} + \frac{1}{19} a^{4} + \frac{6}{19} a^{3} + \frac{6}{19} a^{2} + \frac{2}{19} a$, $\frac{1}{209} a^{9} - \frac{69}{209} a^{7} - \frac{51}{209} a^{6} + \frac{2}{209} a^{5} + \frac{42}{209} a^{4} - \frac{63}{209} a^{3} + \frac{104}{209} a^{2} + \frac{53}{209} a + \frac{4}{11}$, $\frac{1}{209} a^{10} - \frac{3}{209} a^{8} + \frac{81}{209} a^{7} + \frac{101}{209} a^{6} + \frac{9}{209} a^{5} + \frac{3}{209} a^{4} + \frac{82}{209} a^{3} + \frac{31}{209} a^{2} - \frac{1}{209} a$, $\frac{1}{22829470222215888165021121362449} a^{11} + \frac{45780017973093265498271519013}{22829470222215888165021121362449} a^{10} + \frac{46616885901185639470383196520}{22829470222215888165021121362449} a^{9} + \frac{175511980997474625187197755496}{22829470222215888165021121362449} a^{8} - \frac{749886571955321380923570070230}{2075406383837808015001920123859} a^{7} - \frac{7649976640103301816778340793537}{22829470222215888165021121362449} a^{6} + \frac{7438141123713615931674698657144}{22829470222215888165021121362449} a^{5} - \frac{6161147023255820699831689734686}{22829470222215888165021121362449} a^{4} - \frac{1981115067728447088407936952210}{22829470222215888165021121362449} a^{3} + \frac{3384937003651269602903635421079}{22829470222215888165021121362449} a^{2} - \frac{679490270942364175485192452016}{22829470222215888165021121362449} a - \frac{45438065516194889802344872072}{1201551064327152008685322176971}$

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

Class group and class number

$C_{2}\times C_{8770}$, which has order $17540$ (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:  \( 201.000834787 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.595125.1, 6.6.820125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$3.12.18.74$x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$23$23.12.6.2$x^{12} - 6436343 x^{2} + 2220538335$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$