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Properties

Label 12.0.26057506292...8144.3
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{18}\cdot 37^{6}$
Root discriminant $89.40$
Ramified primes $2, 3, 37$
Class number $16128$ (GRH)
Class group $[4, 4, 1008]$ (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![20331721, -2382072, 6626874, -1441552, 924369, -130200, 74498, -4536, 3786, -56, 96, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 96*x^10 - 56*x^9 + 3786*x^8 - 4536*x^7 + 74498*x^6 - 130200*x^5 + 924369*x^4 - 1441552*x^3 + 6626874*x^2 - 2382072*x + 20331721)
 
gp: K = bnfinit(x^12 + 96*x^10 - 56*x^9 + 3786*x^8 - 4536*x^7 + 74498*x^6 - 130200*x^5 + 924369*x^4 - 1441552*x^3 + 6626874*x^2 - 2382072*x + 20331721, 1)
 

Normalized defining polynomial

\( x^{12} + 96 x^{10} - 56 x^{9} + 3786 x^{8} - 4536 x^{7} + 74498 x^{6} - 130200 x^{5} + 924369 x^{4} - 1441552 x^{3} + 6626874 x^{2} - 2382072 x + 20331721 \)

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:  \(260575062921050587398144=2^{18}\cdot 3^{18}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.40$
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:  $2, 3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2664=2^{3}\cdot 3^{2}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2664}(1,·)$, $\chi_{2664}(1553,·)$, $\chi_{2664}(2441,·)$, $\chi_{2664}(1997,·)$, $\chi_{2664}(2221,·)$, $\chi_{2664}(1777,·)$, $\chi_{2664}(1109,·)$, $\chi_{2664}(665,·)$, $\chi_{2664}(889,·)$, $\chi_{2664}(221,·)$, $\chi_{2664}(445,·)$, $\chi_{2664}(1333,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{20} a^{6} + \frac{2}{5} a^{4} + \frac{1}{10} a^{3} + \frac{1}{20} a^{2} + \frac{3}{10} a - \frac{1}{20}$, $\frac{1}{20} a^{7} + \frac{2}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{20} a^{3} + \frac{3}{10} a^{2} - \frac{1}{20} a$, $\frac{1}{20} a^{8} + \frac{1}{10} a^{5} - \frac{3}{20} a^{4} - \frac{1}{2} a^{3} - \frac{9}{20} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{30260} a^{9} - \frac{559}{30260} a^{8} - \frac{523}{30260} a^{7} + \frac{179}{15130} a^{6} - \frac{2089}{6052} a^{5} - \frac{231}{30260} a^{4} - \frac{847}{3026} a^{3} - \frac{2279}{30260} a^{2} - \frac{7661}{30260} a + \frac{2603}{7565}$, $\frac{1}{30260} a^{10} + \frac{11}{1780} a^{8} + \frac{1}{3026} a^{7} + \frac{138}{7565} a^{6} - \frac{458}{7565} a^{5} - \frac{4497}{15130} a^{4} + \frac{161}{1513} a^{3} + \frac{1107}{7565} a^{2} - \frac{1957}{15130} a + \frac{11901}{30260}$, $\frac{1}{99838765255305726515860} a^{11} - \frac{673429891332841741}{49919382627652863257930} a^{10} - \frac{286793381586547455}{19967753051061145303172} a^{9} + \frac{234402825236944682533}{49919382627652863257930} a^{8} - \frac{119482063918526372473}{19967753051061145303172} a^{7} - \frac{993402908320170535253}{49919382627652863257930} a^{6} - \frac{845082546034137449890}{4991938262765286325793} a^{5} - \frac{7699020974055025064966}{24959691313826431628965} a^{4} - \frac{45400793585706197243403}{99838765255305726515860} a^{3} + \frac{7248258642933154822497}{24959691313826431628965} a^{2} - \frac{10426183391883322938413}{49919382627652863257930} a - \frac{3214615395668415803779}{24959691313826431628965}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{1008}$, which has order $16128$ (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:  \( 481.70037561485367 \) (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

\(\Q(\sqrt{-222}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-111}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{2}, \sqrt{-111})\), 6.0.510465535488.3, 6.6.3359232.1, 6.0.997002999.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$