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Properties

Label 12.0.16658028075...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 31^{8}$
Root discriminant $33.00$
Ramified primes $5, 31$
Class number $9$
Class group $[3, 3]$
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![4096, -5120, 6912, -8768, 11184, -632, 1092, -46, 115, -13, 11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 11*x^10 - 13*x^9 + 115*x^8 - 46*x^7 + 1092*x^6 - 632*x^5 + 11184*x^4 - 8768*x^3 + 6912*x^2 - 5120*x + 4096)
 
gp: K = bnfinit(x^12 - x^11 + 11*x^10 - 13*x^9 + 115*x^8 - 46*x^7 + 1092*x^6 - 632*x^5 + 11184*x^4 - 8768*x^3 + 6912*x^2 - 5120*x + 4096, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 11 x^{10} - 13 x^{9} + 115 x^{8} - 46 x^{7} + 1092 x^{6} - 632 x^{5} + 11184 x^{4} - 8768 x^{3} + 6912 x^{2} - 5120 x + 4096 \)

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:  \(1665802807501953125=5^{9}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.00$
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:  $5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(155=5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{155}(32,·)$, $\chi_{155}(1,·)$, $\chi_{155}(98,·)$, $\chi_{155}(67,·)$, $\chi_{155}(36,·)$, $\chi_{155}(129,·)$, $\chi_{155}(149,·)$, $\chi_{155}(118,·)$, $\chi_{155}(87,·)$, $\chi_{155}(56,·)$, $\chi_{155}(94,·)$, $\chi_{155}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{4688704} a^{9} - \frac{123989}{4688704} a^{8} + \frac{37359}{4688704} a^{7} - \frac{105065}{4688704} a^{6} - \frac{513257}{4688704} a^{5} - \frac{975365}{2344352} a^{4} - \frac{275969}{1172176} a^{3} + \frac{101993}{586088} a^{2} - \frac{31281}{293044} a - \frac{2673}{73261}$, $\frac{1}{18754816} a^{10} - \frac{1}{18754816} a^{9} - \frac{22533}{18754816} a^{8} - \frac{476925}{18754816} a^{7} + \frac{251587}{18754816} a^{6} - \frac{1112303}{9377408} a^{5} + \frac{231269}{4688704} a^{4} + \frac{28093}{2344352} a^{3} + \frac{58895}{1172176} a^{2} + \frac{136027}{293044} a + \frac{3210}{73261}$, $\frac{1}{75019264} a^{11} - \frac{1}{75019264} a^{10} - \frac{5}{75019264} a^{9} + \frac{766467}{75019264} a^{8} - \frac{2091581}{75019264} a^{7} + \frac{1677649}{37509632} a^{6} - \frac{4534843}{18754816} a^{5} - \frac{1583027}{9377408} a^{4} - \frac{1651401}{4688704} a^{3} + \frac{311717}{1172176} a^{2} + \frac{15671}{73261} a - \frac{27273}{73261}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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:  \( -\frac{115}{9377408} a^{11} - \frac{353}{153728} a^{6} + \frac{372247}{293044} a \) (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:  \( 4882.16021651 \)
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

\(\Q(\sqrt{5}) \), 3.3.961.1, \(\Q(\zeta_{5})\), 6.6.115440125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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}$$[\ ]_{4}^{3}$
$31$31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.4t1.1c1$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.5.4t1.1c2$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.31.3t1.1c1$1$ $ 31 $ $x^{3} - x^{2} - 10 x + 8$ $C_3$ (as 3T1) $0$ $1$
* 1.5_31.12t1.1c1$1$ $ 5 \cdot 31 $ $x^{12} - x^{11} + 11 x^{10} - 13 x^{9} + 115 x^{8} - 46 x^{7} + 1092 x^{6} - 632 x^{5} + 11184 x^{4} - 8768 x^{3} + 6912 x^{2} - 5120 x + 4096$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_31.6t1.1c1$1$ $ 5 \cdot 31 $ $x^{6} - x^{5} - 24 x^{4} + 7 x^{3} + 112 x^{2} + 23 x - 89$ $C_6$ (as 6T1) $0$ $1$
* 1.5_31.12t1.1c2$1$ $ 5 \cdot 31 $ $x^{12} - x^{11} + 11 x^{10} - 13 x^{9} + 115 x^{8} - 46 x^{7} + 1092 x^{6} - 632 x^{5} + 11184 x^{4} - 8768 x^{3} + 6912 x^{2} - 5120 x + 4096$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.31.3t1.1c2$1$ $ 31 $ $x^{3} - x^{2} - 10 x + 8$ $C_3$ (as 3T1) $0$ $1$
* 1.5_31.12t1.1c3$1$ $ 5 \cdot 31 $ $x^{12} - x^{11} + 11 x^{10} - 13 x^{9} + 115 x^{8} - 46 x^{7} + 1092 x^{6} - 632 x^{5} + 11184 x^{4} - 8768 x^{3} + 6912 x^{2} - 5120 x + 4096$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_31.6t1.1c2$1$ $ 5 \cdot 31 $ $x^{6} - x^{5} - 24 x^{4} + 7 x^{3} + 112 x^{2} + 23 x - 89$ $C_6$ (as 6T1) $0$ $1$
* 1.5_31.12t1.1c4$1$ $ 5 \cdot 31 $ $x^{12} - x^{11} + 11 x^{10} - 13 x^{9} + 115 x^{8} - 46 x^{7} + 1092 x^{6} - 632 x^{5} + 11184 x^{4} - 8768 x^{3} + 6912 x^{2} - 5120 x + 4096$ $C_{12}$ (as 12T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.