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Properties

Label 12.0.13138403152...696.98
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 23^{8}$
Root discriminant $32.35$
Ramified primes $2, 23$
Class number $3$
Class group $[3]$
Galois group $D_6$ (as 12T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16900, 0, -17192, 0, 6092, 0, -1204, 0, 281, 0, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 28*x^10 + 281*x^8 - 1204*x^6 + 6092*x^4 - 17192*x^2 + 16900)
 
gp: K = bnfinit(x^12 - 28*x^10 + 281*x^8 - 1204*x^6 + 6092*x^4 - 17192*x^2 + 16900, 1)
 

Normalized defining polynomial

\( x^{12} - 28 x^{10} + 281 x^{8} - 1204 x^{6} + 6092 x^{4} - 17192 x^{2} + 16900 \)

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:  \(1313840315232157696=2^{24}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.35$
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:  $2, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{88} a^{8} - \frac{7}{44} a^{6} - \frac{1}{2} a^{5} + \frac{43}{88} a^{4} - \frac{7}{44} a^{2} - \frac{1}{2} a + \frac{21}{44}$, $\frac{1}{440} a^{9} - \frac{51}{220} a^{7} - \frac{1}{2} a^{6} + \frac{219}{440} a^{5} - \frac{19}{44} a^{3} - \frac{1}{2} a^{2} + \frac{21}{220} a$, $\frac{1}{138127000} a^{10} + \frac{114913}{138127000} a^{8} - \frac{1}{2} a^{7} + \frac{2960289}{138127000} a^{6} - \frac{1}{2} a^{5} - \frac{5341101}{27625400} a^{4} - \frac{1}{2} a^{3} + \frac{6625439}{17265875} a^{2} - \frac{1}{2} a - \frac{625753}{2762540}$, $\frac{1}{3591302000} a^{11} + \frac{842269}{1795651000} a^{9} + \frac{1362255539}{3591302000} a^{7} - \frac{1}{2} a^{6} - \frac{37359263}{359130200} a^{5} - \frac{1}{2} a^{4} - \frac{329803119}{1795651000} a^{3} - \frac{1}{2} a^{2} - \frac{3711901}{8978255} a - \frac{1}{2}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{447687}{1795651000} a^{11} + \frac{5666147}{897825500} a^{9} - \frac{94347093}{1795651000} a^{7} + \frac{25897791}{179565100} a^{5} - \frac{875051747}{897825500} a^{3} + \frac{10673288}{8978255} a \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 103831.352805 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_6$ (as 12T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $D_6$
Character table for $D_6$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), 3.1.4232.1 x3, \(\Q(\zeta_{8})\), 6.0.143278592.1, 6.2.573114368.1 x3, 6.0.286557184.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
$23$23.6.4.1$x^{6} + 460 x^{3} + 181447$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
23.6.4.1$x^{6} + 460 x^{3} + 181447$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$