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Properties

Label 12.0.13179165217...0000.9
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 5^{6}\cdot 7^{10}$
Root discriminant $39.20$
Ramified primes $2, 3, 5, 7$
Class number $104$
Class group $[2, 52]$
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28561, 10088, 5381, -5206, 4243, -390, 20, -204, 100, -2, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - x^10 - 2*x^9 + 100*x^8 - 204*x^7 + 20*x^6 - 390*x^5 + 4243*x^4 - 5206*x^3 + 5381*x^2 + 10088*x + 28561)
 
gp: K = bnfinit(x^12 - 2*x^11 - x^10 - 2*x^9 + 100*x^8 - 204*x^7 + 20*x^6 - 390*x^5 + 4243*x^4 - 5206*x^3 + 5381*x^2 + 10088*x + 28561, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - x^{10} - 2 x^{9} + 100 x^{8} - 204 x^{7} + 20 x^{6} - 390 x^{5} + 4243 x^{4} - 5206 x^{3} + 5381 x^{2} + 10088 x + 28561 \)

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:  \(13179165217344000000=2^{12}\cdot 3^{6}\cdot 5^{6}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.20$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(419,·)$, $\chi_{420}(229,·)$, $\chi_{420}(71,·)$, $\chi_{420}(361,·)$, $\chi_{420}(11,·)$, $\chi_{420}(409,·)$, $\chi_{420}(121,·)$, $\chi_{420}(59,·)$, $\chi_{420}(349,·)$, $\chi_{420}(299,·)$, $\chi_{420}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13} a^{7} + \frac{5}{13} a^{6} - \frac{4}{13} a^{5} + \frac{1}{13} a^{3} - \frac{6}{13} a^{2} + \frac{1}{13} a$, $\frac{1}{13} a^{8} - \frac{3}{13} a^{6} - \frac{6}{13} a^{5} + \frac{1}{13} a^{4} + \frac{2}{13} a^{3} + \frac{5}{13} a^{2} - \frac{5}{13} a$, $\frac{1}{13} a^{9} - \frac{4}{13} a^{6} + \frac{2}{13} a^{5} + \frac{2}{13} a^{4} - \frac{5}{13} a^{3} + \frac{3}{13} a^{2} + \frac{3}{13} a$, $\frac{1}{13} a^{10} - \frac{4}{13} a^{6} - \frac{1}{13} a^{5} - \frac{5}{13} a^{4} - \frac{6}{13} a^{3} + \frac{5}{13} a^{2} + \frac{4}{13} a$, $\frac{1}{3909538304231901263} a^{11} + \frac{31452195896210550}{3909538304231901263} a^{10} - \frac{113298089534671358}{3909538304231901263} a^{9} + \frac{131565216238518553}{3909538304231901263} a^{8} + \frac{143776824973202718}{3909538304231901263} a^{7} + \frac{1233503530343349123}{3909538304231901263} a^{6} - \frac{1341432369704671606}{3909538304231901263} a^{5} + \frac{401901373297437666}{3909538304231901263} a^{4} - \frac{1565033834234311698}{3909538304231901263} a^{3} + \frac{1894167432124528072}{3909538304231901263} a^{2} - \frac{4334063878617632}{23133362746934327} a - \frac{3518207385579834}{23133362746934327}$

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

Class group and class number

$C_{2}\times C_{52}$, which has order $104$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 553.066702068 \)
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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{-35}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{3}, \sqrt{-35})\), 6.6.4148928.1, 6.0.3630312000.2, 6.0.2100875.1

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 R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$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$$[\ ]_{2}^{3}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$