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Properties

Label 12.0.17318861576...5669.1
Degree $12$
Signature $[0, 6]$
Discriminant $229^{9}$
Root discriminant $58.87$
Ramified prime $229$
Class number $272$ (GRH)
Class group $[2, 2, 2, 34]$ (GRH)
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![105011, -134592, 73305, 14489, 23869, -10397, 4875, -1078, 433, -122, 54, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^11 + 54*x^10 - 122*x^9 + 433*x^8 - 1078*x^7 + 4875*x^6 - 10397*x^5 + 23869*x^4 + 14489*x^3 + 73305*x^2 - 134592*x + 105011)
 
gp: K = bnfinit(x^12 - 5*x^11 + 54*x^10 - 122*x^9 + 433*x^8 - 1078*x^7 + 4875*x^6 - 10397*x^5 + 23869*x^4 + 14489*x^3 + 73305*x^2 - 134592*x + 105011, 1)
 

Normalized defining polynomial

\( x^{12} - 5 x^{11} + 54 x^{10} - 122 x^{9} + 433 x^{8} - 1078 x^{7} + 4875 x^{6} - 10397 x^{5} + 23869 x^{4} + 14489 x^{3} + 73305 x^{2} - 134592 x + 105011 \)

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:  \(1731886157602686265669=229^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.87$
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:  $229$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{106} a^{8} + \frac{23}{106} a^{7} + \frac{11}{53} a^{6} - \frac{1}{106} a^{5} + \frac{37}{106} a^{4} + \frac{31}{106} a^{3} - \frac{16}{53} a^{2} - \frac{1}{2} a + \frac{25}{106}$, $\frac{1}{106} a^{9} + \frac{23}{106} a^{7} + \frac{23}{106} a^{6} - \frac{23}{53} a^{5} + \frac{14}{53} a^{4} - \frac{3}{106} a^{3} + \frac{47}{106} a^{2} - \frac{14}{53} a - \frac{45}{106}$, $\frac{1}{318} a^{10} - \frac{1}{318} a^{8} + \frac{107}{318} a^{7} - \frac{25}{53} a^{6} - \frac{9}{53} a^{5} + \frac{21}{106} a^{4} + \frac{151}{318} a^{3} - \frac{18}{53} a^{2} + \frac{61}{318} a - \frac{35}{159}$, $\frac{1}{108075189242400010505802} a^{11} + \frac{80237512290039351097}{54037594621200005252901} a^{10} - \frac{268014039856919968531}{108075189242400010505802} a^{9} - \frac{49200195742613718893}{18012531540400001750967} a^{8} - \frac{250435978358968462151}{108075189242400010505802} a^{7} + \frac{2763592669429093857983}{18012531540400001750967} a^{6} + \frac{2903089230973532315869}{6004177180133333916989} a^{5} + \frac{26443537461058505036723}{54037594621200005252901} a^{4} + \frac{45923441926826818809155}{108075189242400010505802} a^{3} + \frac{8268971934520702097215}{108075189242400010505802} a^{2} - \frac{976472510746221800441}{108075189242400010505802} a - \frac{34631042633264622616319}{108075189242400010505802}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{34}$, which has order $272$ (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:  \( 1444.72403292 \) (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

\(\Q(\sqrt{229}) \), 3.3.229.1 x3, 4.0.12008989.1, 6.6.12008989.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
229Data not computed