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Properties

Label 12.0.68603114334...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 37^{8}$
Root discriminant $37.13$
Ramified primes $5, 37$
Class number $49$
Class group $[7, 7]$
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![14641, 15972, 16093, 14773, 13201, 4269, 2114, 718, 203, -36, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 13*x^10 - 36*x^9 + 203*x^8 + 718*x^7 + 2114*x^6 + 4269*x^5 + 13201*x^4 + 14773*x^3 + 16093*x^2 + 15972*x + 14641)
 
gp: K = bnfinit(x^12 - x^11 + 13*x^10 - 36*x^9 + 203*x^8 + 718*x^7 + 2114*x^6 + 4269*x^5 + 13201*x^4 + 14773*x^3 + 16093*x^2 + 15972*x + 14641, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 13 x^{10} - 36 x^{9} + 203 x^{8} + 718 x^{7} + 2114 x^{6} + 4269 x^{5} + 13201 x^{4} + 14773 x^{3} + 16093 x^{2} + 15972 x + 14641 \)

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:  \(6860311433439453125=5^{9}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.13$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(185=5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{185}(1,·)$, $\chi_{185}(38,·)$, $\chi_{185}(137,·)$, $\chi_{185}(174,·)$, $\chi_{185}(47,·)$, $\chi_{185}(112,·)$, $\chi_{185}(84,·)$, $\chi_{185}(149,·)$, $\chi_{185}(121,·)$, $\chi_{185}(26,·)$, $\chi_{185}(158,·)$, $\chi_{185}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{35327941} a^{9} - \frac{5263512}{35327941} a^{8} - \frac{7788669}{35327941} a^{7} + \frac{15282396}{35327941} a^{6} - \frac{15519851}{35327941} a^{5} - \frac{8653928}{35327941} a^{4} + \frac{11686358}{35327941} a^{3} + \frac{10288950}{35327941} a^{2} + \frac{13172908}{35327941} a + \frac{1552045}{3211631}$, $\frac{1}{388607351} a^{10} - \frac{1}{388607351} a^{9} + \frac{140152014}{388607351} a^{8} - \frac{162427774}{388607351} a^{7} - \frac{98784802}{388607351} a^{6} - \frac{187792371}{388607351} a^{5} + \frac{11865559}{388607351} a^{4} - \frac{172267908}{388607351} a^{3} + \frac{82848349}{388607351} a^{2} + \frac{4641792}{35327941} a + \frac{743736}{3211631}$, $\frac{1}{4274680861} a^{11} - \frac{1}{4274680861} a^{10} + \frac{13}{4274680861} a^{9} + \frac{573900544}{4274680861} a^{8} + \frac{563554920}{4274680861} a^{7} + \frac{978118012}{4274680861} a^{6} - \frac{528364956}{4274680861} a^{5} + \frac{1791996119}{4274680861} a^{4} + \frac{913079201}{4274680861} a^{3} + \frac{6904272}{388607351} a^{2} - \frac{2557158}{35327941} a - \frac{205657}{3211631}$

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

Class group and class number

$C_{7}\times C_{7}$, which has order $49$

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:  \( \frac{203}{35327941} a^{11} + \frac{26265}{3211631} a^{6} - \frac{31119434}{35327941} a \) (order $10$)
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:  \( 5133.82158211 \)
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

\(\Q(\sqrt{5}) \), 3.3.1369.1, \(\Q(\zeta_{5})\), 6.6.234270125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$37$37.12.8.1$x^{12} - 111 x^{9} + 4107 x^{6} - 50653 x^{3} + 14993288$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$