↑

Properties

Label 12.0.22197040583...0409.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{10}\cdot 47^{6}$
Root discriminant $60.10$
Ramified primes $3, 7, 47$
Class number $1860$ (GRH)
Class group $[2, 930]$ (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![7035133, -1175923, 2412291, -412622, 379456, -60340, 35085, -4720, 1985, -204, 65, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 65*x^10 - 204*x^9 + 1985*x^8 - 4720*x^7 + 35085*x^6 - 60340*x^5 + 379456*x^4 - 412622*x^3 + 2412291*x^2 - 1175923*x + 7035133)
 
gp: K = bnfinit(x^12 - 4*x^11 + 65*x^10 - 204*x^9 + 1985*x^8 - 4720*x^7 + 35085*x^6 - 60340*x^5 + 379456*x^4 - 412622*x^3 + 2412291*x^2 - 1175923*x + 7035133, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 65 x^{10} - 204 x^{9} + 1985 x^{8} - 4720 x^{7} + 35085 x^{6} - 60340 x^{5} + 379456 x^{4} - 412622 x^{3} + 2412291 x^{2} - 1175923 x + 7035133 \)

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:  \(2219704058347157210409=3^{6}\cdot 7^{10}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.10$
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:  $3, 7, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(987=3\cdot 7\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{987}(1,·)$, $\chi_{987}(610,·)$, $\chi_{987}(424,·)$, $\chi_{987}(236,·)$, $\chi_{987}(941,·)$, $\chi_{987}(142,·)$, $\chi_{987}(845,·)$, $\chi_{987}(563,·)$, $\chi_{987}(46,·)$, $\chi_{987}(377,·)$, $\chi_{987}(986,·)$, $\chi_{987}(751,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{53652373476957795446310347} a^{11} - \frac{17218448569836269729193840}{53652373476957795446310347} a^{10} - \frac{25697616363763088101809997}{53652373476957795446310347} a^{9} - \frac{16753152944673108166752254}{53652373476957795446310347} a^{8} + \frac{1413791548629868599648176}{53652373476957795446310347} a^{7} - \frac{15914656894288223780856291}{53652373476957795446310347} a^{6} + \frac{15525425324904807699531332}{53652373476957795446310347} a^{5} - \frac{7475522732581469410250881}{53652373476957795446310347} a^{4} + \frac{14598530964427752624768063}{53652373476957795446310347} a^{3} + \frac{17815389763026859029601751}{53652373476957795446310347} a^{2} - \frac{25160391121320649510055876}{53652373476957795446310347} a - \frac{312713575354681766705107}{53652373476957795446310347}$

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

Class group and class number

$C_{2}\times C_{930}$, which has order $1860$ (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:  \( 140.7987960054707 \) (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

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-987}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-47})\), 6.0.249279023.2, \(\Q(\zeta_{21})^+\), 6.0.47113735347.2

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$47$47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$