↑

Properties

Label 12.12.5620365587...9941.2
Degree $12$
Signature $[12, 0]$
Discriminant $3^{18}\cdot 29^{9}$
Root discriminant $64.94$
Ramified primes $3, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3 : C_4$ (as 12T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21625, 83475, 116550, 57737, -15819, -25443, -4810, 2802, 990, -92, -57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 57*x^10 - 92*x^9 + 990*x^8 + 2802*x^7 - 4810*x^6 - 25443*x^5 - 15819*x^4 + 57737*x^3 + 116550*x^2 + 83475*x + 21625)
 
gp: K = bnfinit(x^12 - 57*x^10 - 92*x^9 + 990*x^8 + 2802*x^7 - 4810*x^6 - 25443*x^5 - 15819*x^4 + 57737*x^3 + 116550*x^2 + 83475*x + 21625, 1)
 

Normalized defining polynomial

\( x^{12} - 57 x^{10} - 92 x^{9} + 990 x^{8} + 2802 x^{7} - 4810 x^{6} - 25443 x^{5} - 15819 x^{4} + 57737 x^{3} + 116550 x^{2} + 83475 x + 21625 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5620365587965550179941=3^{18}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.94$
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:  $3, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{9} + \frac{3}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{2} a^{6} - \frac{3}{10} a^{5} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2} + \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{3852527050} a^{11} - \frac{3710923}{770505410} a^{10} - \frac{1563234507}{3852527050} a^{9} - \frac{675183906}{1926263525} a^{8} - \frac{21580501}{770505410} a^{7} + \frac{475415277}{3852527050} a^{6} + \frac{109289856}{385252705} a^{5} + \frac{770159316}{1926263525} a^{4} - \frac{1043352899}{3852527050} a^{3} + \frac{641726911}{1926263525} a^{2} + \frac{366625259}{770505410} a - \frac{13753101}{77050541}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 6418973.82526 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

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 $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), 3.3.2349.1 x3, 4.4.219501.1, 6.6.160016229.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.12.18.68$x^{12} + 21 x^{11} - 21 x^{10} + 21 x^{9} - 27 x^{7} + 15 x^{6} + 18 x^{5} - 27 x^{4} + 27 x^{3} + 27 x^{2} + 27 x - 36$$6$$2$$18$$C_3 : C_4$$[2]_{2}^{2}$
$29$29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$