↑

Properties

Label 12.0.81564235816...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 5^{9}\cdot 47^{6}$
Root discriminant $119.11$
Ramified primes $3, 5, 47$
Class number $136404$ (GRH)
Class group $[18, 7578]$ (GRH)
Galois group $C_{12}$ (as 12T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4170430441, -700201812, 763654464, -55917689, 60795645, -1132947, 2316848, -6651, 42174, -1, 348, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 348*x^10 - x^9 + 42174*x^8 - 6651*x^7 + 2316848*x^6 - 1132947*x^5 + 60795645*x^4 - 55917689*x^3 + 763654464*x^2 - 700201812*x + 4170430441)
 
gp: K = bnfinit(x^12 + 348*x^10 - x^9 + 42174*x^8 - 6651*x^7 + 2316848*x^6 - 1132947*x^5 + 60795645*x^4 - 55917689*x^3 + 763654464*x^2 - 700201812*x + 4170430441, 1)
 

Normalized defining polynomial

\( x^{12} + 348 x^{10} - x^{9} + 42174 x^{8} - 6651 x^{7} + 2316848 x^{6} - 1132947 x^{5} + 60795645 x^{4} - 55917689 x^{3} + 763654464 x^{2} - 700201812 x + 4170430441 \)

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:  \(8156423581635695080078125=3^{18}\cdot 5^{9}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $119.11$
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:  $3, 5, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2115=3^{2}\cdot 5\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{2115}(1,·)$, $\chi_{2115}(706,·)$, $\chi_{2115}(1411,·)$, $\chi_{2115}(422,·)$, $\chi_{2115}(1127,·)$, $\chi_{2115}(1832,·)$, $\chi_{2115}(1129,·)$, $\chi_{2115}(1834,·)$, $\chi_{2115}(424,·)$, $\chi_{2115}(563,·)$, $\chi_{2115}(1268,·)$, $\chi_{2115}(1973,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{19} a^{8} - \frac{4}{19} a^{7} + \frac{2}{19} a^{6} - \frac{1}{19} a^{5} - \frac{7}{19} a^{4} + \frac{3}{19} a^{3} + \frac{5}{19} a^{2} - \frac{3}{19} a$, $\frac{1}{779} a^{9} + \frac{290}{779} a^{7} - \frac{221}{779} a^{6} - \frac{353}{779} a^{5} + \frac{13}{779} a^{4} - \frac{135}{779} a^{3} + \frac{283}{779} a^{2} - \frac{335}{779} a - \frac{13}{41}$, $\frac{1}{779} a^{10} + \frac{3}{779} a^{8} + \frac{148}{779} a^{7} - \frac{148}{779} a^{6} + \frac{300}{779} a^{5} + \frac{316}{779} a^{4} + \frac{201}{779} a^{3} - \frac{212}{779} a^{2} - \frac{165}{779} a$, $\frac{1}{206151888274470543026001775681166354069} a^{11} - \frac{98776138002273753081998835126645930}{206151888274470543026001775681166354069} a^{10} + \frac{61829793171706463150581749205918404}{206151888274470543026001775681166354069} a^{9} + \frac{4529510225495091031810958615168139208}{206151888274470543026001775681166354069} a^{8} + \frac{38536874533757805841127321281565089182}{206151888274470543026001775681166354069} a^{7} + \frac{88641012491150989422265754923200792944}{206151888274470543026001775681166354069} a^{6} - \frac{45594665550383911347190627255287485888}{206151888274470543026001775681166354069} a^{5} + \frac{28489866576515928299037850678859540217}{206151888274470543026001775681166354069} a^{4} + \frac{2505429191956571186657248523038373024}{10850099382866870685579040825324544951} a^{3} - \frac{91539778258608432098197501883679553191}{206151888274470543026001775681166354069} a^{2} - \frac{75436829910627718237990604366053177383}{206151888274470543026001775681166354069} a + \frac{1074771884039343114391583978000719465}{10850099382866870685579040825324544951}$

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

Class group and class number

$C_{18}\times C_{7578}$, which has order $136404$ (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:  \( 201.000834787 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

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

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.2485125.1, 6.6.820125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$3.12.18.74$x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$47$47.12.6.2$x^{12} - 229345007 x^{2} + 53896076645$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$