↑

Properties

Label 12.12.2951578112...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{18}\cdot 5^{9}\cdot 7^{8}$
Root discriminant $34.61$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (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![-239, 2052, -1580, -6484, 2480, 4864, -1458, -1252, 339, 124, -32, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 32*x^10 + 124*x^9 + 339*x^8 - 1252*x^7 - 1458*x^6 + 4864*x^5 + 2480*x^4 - 6484*x^3 - 1580*x^2 + 2052*x - 239)
 
gp: K = bnfinit(x^12 - 4*x^11 - 32*x^10 + 124*x^9 + 339*x^8 - 1252*x^7 - 1458*x^6 + 4864*x^5 + 2480*x^4 - 6484*x^3 - 1580*x^2 + 2052*x - 239, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 32 x^{10} + 124 x^{9} + 339 x^{8} - 1252 x^{7} - 1458 x^{6} + 4864 x^{5} + 2480 x^{4} - 6484 x^{3} - 1580 x^{2} + 2052 x - 239 \)

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:  \(2951578112000000000=2^{18}\cdot 5^{9}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.61$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(280=2^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{280}(107,·)$, $\chi_{280}(1,·)$, $\chi_{280}(67,·)$, $\chi_{280}(9,·)$, $\chi_{280}(267,·)$, $\chi_{280}(81,·)$, $\chi_{280}(163,·)$, $\chi_{280}(43,·)$, $\chi_{280}(121,·)$, $\chi_{280}(249,·)$, $\chi_{280}(123,·)$, $\chi_{280}(169,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{62078} a^{10} - \frac{2752}{31039} a^{9} - \frac{3793}{31039} a^{8} + \frac{7015}{62078} a^{7} - \frac{4625}{31039} a^{6} - \frac{19443}{62078} a^{5} - \frac{15108}{31039} a^{4} - \frac{12720}{31039} a^{3} - \frac{9617}{62078} a^{2} + \frac{8987}{62078} a + \frac{23721}{62078}$, $\frac{1}{114548870798} a^{11} + \frac{91610}{57274435399} a^{10} + \frac{339356079}{114548870798} a^{9} + \frac{3991356675}{114548870798} a^{8} + \frac{229671743}{114548870798} a^{7} - \frac{11731433345}{57274435399} a^{6} - \frac{2045884830}{57274435399} a^{5} - \frac{48958056717}{114548870798} a^{4} + \frac{963044744}{57274435399} a^{3} - \frac{51946411089}{114548870798} a^{2} + \frac{27099397157}{114548870798} a + \frac{6993022695}{114548870798}$

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:  \( 193697.65851 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.4.8000.1, 6.6.300125.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$