↑

Properties

Label 12.0.17873674734...6064.3
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 7^{10}\cdot 17^{6}$
Root discriminant $59.02$
Ramified primes $2, 7, 17$
Class number $1720$ (GRH)
Class group $[2, 2, 430]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1573111, -229794, 663283, -198676, 133099, -30618, 14855, -2310, 1107, -96, 47, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 47*x^10 - 96*x^9 + 1107*x^8 - 2310*x^7 + 14855*x^6 - 30618*x^5 + 133099*x^4 - 198676*x^3 + 663283*x^2 - 229794*x + 1573111)
 
gp: K = bnfinit(x^12 - 2*x^11 + 47*x^10 - 96*x^9 + 1107*x^8 - 2310*x^7 + 14855*x^6 - 30618*x^5 + 133099*x^4 - 198676*x^3 + 663283*x^2 - 229794*x + 1573111, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 47 x^{10} - 96 x^{9} + 1107 x^{8} - 2310 x^{7} + 14855 x^{6} - 30618 x^{5} + 133099 x^{4} - 198676 x^{3} + 663283 x^{2} - 229794 x + 1573111 \)

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:  \(1787367473421924696064=2^{18}\cdot 7^{10}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.02$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(952=2^{3}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{952}(1,·)$, $\chi_{952}(477,·)$, $\chi_{952}(101,·)$, $\chi_{952}(577,·)$, $\chi_{952}(137,·)$, $\chi_{952}(205,·)$, $\chi_{952}(681,·)$, $\chi_{952}(237,·)$, $\chi_{952}(713,·)$, $\chi_{952}(33,·)$, $\chi_{952}(509,·)$, $\chi_{952}(613,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{181} a^{8} - \frac{76}{181} a^{7} + \frac{20}{181} a^{6} - \frac{43}{181} a^{5} + \frac{27}{181} a^{4} - \frac{68}{181} a^{3} - \frac{16}{181} a^{2} + \frac{60}{181} a + \frac{38}{181}$, $\frac{1}{181} a^{9} + \frac{36}{181} a^{7} + \frac{29}{181} a^{6} + \frac{17}{181} a^{5} - \frac{7}{181} a^{4} + \frac{65}{181} a^{3} - \frac{70}{181} a^{2} + \frac{73}{181} a - \frac{8}{181}$, $\frac{1}{7421} a^{10} - \frac{5}{7421} a^{9} - \frac{2}{7421} a^{8} + \frac{927}{7421} a^{7} + \frac{3456}{7421} a^{6} + \frac{3352}{7421} a^{5} + \frac{341}{7421} a^{4} + \frac{3275}{7421} a^{3} + \frac{1574}{7421} a^{2} - \frac{2834}{7421} a - \frac{3033}{7421}$, $\frac{1}{2168939639047245076643} a^{11} - \frac{104068034795338699}{2168939639047245076643} a^{10} + \frac{3862099248472341628}{2168939639047245076643} a^{9} - \frac{3452935126408114098}{2168939639047245076643} a^{8} + \frac{220331699314883318227}{2168939639047245076643} a^{7} - \frac{323305565742519055275}{2168939639047245076643} a^{6} - \frac{470566235393441600405}{2168939639047245076643} a^{5} - \frac{775290896747360465874}{2168939639047245076643} a^{4} + \frac{233356961843416337663}{2168939639047245076643} a^{3} - \frac{19514263397599885871}{52900966806030367723} a^{2} + \frac{117598370431142093389}{2168939639047245076643} a + \frac{954853527833401104435}{2168939639047245076643}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{430}$, which has order $1720$ (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:  \( 279.1500271937239 \) (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

\(\Q(\sqrt{-238}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-119}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-119})\), 6.0.42277268992.6, 6.6.1229312.1, 6.0.82572791.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$