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Properties

Label 12.0.22039921152...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{16}\cdot 5^{9}$
Root discriminant $40.92$
Ramified primes $2, 3, 5$
Class number $218$ (GRH)
Class group $[218]$ (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![70471, 21732, 40764, 3624, 8355, 72, 1738, 36, 234, -4, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 18*x^10 - 4*x^9 + 234*x^8 + 36*x^7 + 1738*x^6 + 72*x^5 + 8355*x^4 + 3624*x^3 + 40764*x^2 + 21732*x + 70471)
 
gp: K = bnfinit(x^12 + 18*x^10 - 4*x^9 + 234*x^8 + 36*x^7 + 1738*x^6 + 72*x^5 + 8355*x^4 + 3624*x^3 + 40764*x^2 + 21732*x + 70471, 1)
 

Normalized defining polynomial

\( x^{12} + 18 x^{10} - 4 x^{9} + 234 x^{8} + 36 x^{7} + 1738 x^{6} + 72 x^{5} + 8355 x^{4} + 3624 x^{3} + 40764 x^{2} + 21732 x + 70471 \)

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:  \(22039921152000000000=2^{18}\cdot 3^{16}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.92$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(360=2^{3}\cdot 3^{2}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{360}(1,·)$, $\chi_{360}(133,·)$, $\chi_{360}(289,·)$, $\chi_{360}(169,·)$, $\chi_{360}(13,·)$, $\chi_{360}(157,·)$, $\chi_{360}(241,·)$, $\chi_{360}(277,·)$, $\chi_{360}(121,·)$, $\chi_{360}(49,·)$, $\chi_{360}(253,·)$, $\chi_{360}(37,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{5585362} a^{10} + \frac{758165}{5585362} a^{9} - \frac{427738}{2792681} a^{8} + \frac{239976}{2792681} a^{7} + \frac{354430}{2792681} a^{6} + \frac{2316223}{5585362} a^{5} + \frac{2117503}{5585362} a^{4} + \frac{1318169}{5585362} a^{3} - \frac{95705}{5585362} a^{2} + \frac{123477}{2792681} a - \frac{1261353}{5585362}$, $\frac{1}{1055974130667362} a^{11} + \frac{40017351}{527987065333681} a^{10} - \frac{69671974571546}{527987065333681} a^{9} - \frac{91393683106633}{1055974130667362} a^{8} - \frac{67819743910533}{527987065333681} a^{7} - \frac{257384110748271}{1055974130667362} a^{6} + \frac{248973933862949}{1055974130667362} a^{5} + \frac{223094368934365}{1055974130667362} a^{4} - \frac{9656623234339}{1055974130667362} a^{3} - \frac{3027646338729}{7436437539911} a^{2} + \frac{143604758620187}{1055974130667362} a - \frac{11966175739309}{55577585824598}$

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

Class group and class number

$C_{218}$, which has order $218$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.8000.2, 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 R R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$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}$$[3]^{6}$
$3$3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$