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Properties

Label 12.0.47832530538...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 5^{9}\cdot 43^{6}$
Root discriminant $113.93$
Ramified primes $3, 5, 43$
Class number $104180$ (GRH)
Class group $[2, 52090]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2516315671, -445296567, 490256874, -39182024, 42372270, -870237, 1766048, -5601, 35199, -1, 318, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 318*x^10 - x^9 + 35199*x^8 - 5601*x^7 + 1766048*x^6 - 870237*x^5 + 42372270*x^4 - 39182024*x^3 + 490256874*x^2 - 445296567*x + 2516315671)
 
gp: K = bnfinit(x^12 + 318*x^10 - x^9 + 35199*x^8 - 5601*x^7 + 1766048*x^6 - 870237*x^5 + 42372270*x^4 - 39182024*x^3 + 490256874*x^2 - 445296567*x + 2516315671, 1)
 

Normalized defining polynomial

\( x^{12} + 318 x^{10} - x^{9} + 35199 x^{8} - 5601 x^{7} + 1766048 x^{6} - 870237 x^{5} + 42372270 x^{4} - 39182024 x^{3} + 490256874 x^{2} - 445296567 x + 2516315671 \)

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:  \(4783253053886935470703125=3^{18}\cdot 5^{9}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.93$
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:  $3, 5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1935=3^{2}\cdot 5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1935}(128,·)$, $\chi_{1935}(1,·)$, $\chi_{1935}(259,·)$, $\chi_{1935}(773,·)$, $\chi_{1935}(646,·)$, $\chi_{1935}(257,·)$, $\chi_{1935}(904,·)$, $\chi_{1935}(1418,·)$, $\chi_{1935}(1547,·)$, $\chi_{1935}(1549,·)$, $\chi_{1935}(902,·)$, $\chi_{1935}(1291,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{659} a^{9} + \frac{265}{659} a^{7} + \frac{255}{659} a^{6} + \frac{66}{659} a^{5} - \frac{42}{659} a^{4} - \frac{236}{659} a^{3} + \frac{313}{659} a^{2} + \frac{326}{659} a + \frac{237}{659}$, $\frac{1}{12521} a^{10} - \frac{4}{12521} a^{9} + \frac{3560}{12521} a^{8} - \frac{146}{12521} a^{7} + \frac{1023}{12521} a^{6} + \frac{353}{12521} a^{5} + \frac{4545}{12521} a^{4} + \frac{3893}{12521} a^{3} + \frac{3687}{12521} a^{2} - \frac{4362}{12521} a + \frac{370}{12521}$, $\frac{1}{26869993873716736701279424839687963059} a^{11} - \frac{372958158231288234718113935868728}{26869993873716736701279424839687963059} a^{10} - \frac{19057846066478132359930624983385101}{26869993873716736701279424839687963059} a^{9} + \frac{10220165601756510608267576900869173367}{26869993873716736701279424839687963059} a^{8} - \frac{11969140875301135806677581252541233366}{26869993873716736701279424839687963059} a^{7} - \frac{8823321752116181034168025877771077932}{26869993873716736701279424839687963059} a^{6} + \frac{3889472467661694055257177635923887819}{26869993873716736701279424839687963059} a^{5} + \frac{6315674209561989458361913863521184812}{26869993873716736701279424839687963059} a^{4} + \frac{5494905673246366606040837573039042246}{26869993873716736701279424839687963059} a^{3} + \frac{11827840898856597022000204261412749755}{26869993873716736701279424839687963059} a^{2} - \frac{2031685596235803431937817528811670460}{26869993873716736701279424839687963059} a + \frac{12447028019801171229299565327462851650}{26869993873716736701279424839687963059}$

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

Class group and class number

$C_{2}\times C_{52090}$, which has order $104180$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.2080125.1, 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 ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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}$
$43$43.12.6.2$x^{12} - 147008443 x^{2} + 164355439274$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$