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Properties

Label 12.0.36533490045...296.10
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 13^{10}\cdot 31^{6}$
Root discriminant $163.51$
Ramified primes $2, 3, 13, 31$
Class number $1326080$ (GRH)
Class group $[2, 2, 4, 8, 10360]$ (GRH)
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![726950311489, -14988666216, 44414596734, -767614604, 1146467101, -15988294, 16003958, -169318, 127439, -912, 549, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 549*x^10 - 912*x^9 + 127439*x^8 - 169318*x^7 + 16003958*x^6 - 15988294*x^5 + 1146467101*x^4 - 767614604*x^3 + 44414596734*x^2 - 14988666216*x + 726950311489)
 
gp: K = bnfinit(x^12 - 2*x^11 + 549*x^10 - 912*x^9 + 127439*x^8 - 169318*x^7 + 16003958*x^6 - 15988294*x^5 + 1146467101*x^4 - 767614604*x^3 + 44414596734*x^2 - 14988666216*x + 726950311489, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 549 x^{10} - 912 x^{9} + 127439 x^{8} - 169318 x^{7} + 16003958 x^{6} - 15988294 x^{5} + 1146467101 x^{4} - 767614604 x^{3} + 44414596734 x^{2} - 14988666216 x + 726950311489 \)

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:  \(365334900454961075024695296=2^{12}\cdot 3^{6}\cdot 13^{10}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.51$
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:  $2, 3, 13, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4836=2^{2}\cdot 3\cdot 13\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{4836}(1,·)$, $\chi_{4836}(4835,·)$, $\chi_{4836}(3719,·)$, $\chi_{4836}(745,·)$, $\chi_{4836}(2603,·)$, $\chi_{4836}(4463,·)$, $\chi_{4836}(1115,·)$, $\chi_{4836}(373,·)$, $\chi_{4836}(3721,·)$, $\chi_{4836}(2233,·)$, $\chi_{4836}(4091,·)$, $\chi_{4836}(1117,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{85948340259796119606241918949001290945} a^{11} + \frac{1149547678432931332817530403922069719}{17189668051959223921248383789800258189} a^{10} + \frac{3492462788706833762378333806901356420}{17189668051959223921248383789800258189} a^{9} + \frac{6305099222649211623871127980049480347}{17189668051959223921248383789800258189} a^{8} - \frac{4569836698516848729998747460445571692}{85948340259796119606241918949001290945} a^{7} - \frac{36679932647769526779318886489537231734}{85948340259796119606241918949001290945} a^{6} + \frac{6435766640372145849681320100964302349}{85948340259796119606241918949001290945} a^{5} - \frac{13961320192566173572308931839508758047}{85948340259796119606241918949001290945} a^{4} - \frac{8149041949352881018113185988829097701}{85948340259796119606241918949001290945} a^{3} - \frac{1870738496384970607383726729149067134}{85948340259796119606241918949001290945} a^{2} - \frac{20401038970220403143591585453159432426}{85948340259796119606241918949001290945} a - \frac{2057797858712330146569032902588322}{12379135857669036382866472554947615}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{8}\times C_{10360}$, which has order $1326080$ (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:  \( 120.78403136265631 \) (assuming GRH)
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

\(\Q(\sqrt{-1209}) \), \(\Q(\sqrt{-93}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-93})\), 6.0.19113735910464.2, 6.0.1470287377728.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.12.26$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$$[2]^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$