Properties

Label 12.0.33171021564453125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 19^{8}$
Root discriminant $23.81$
Ramified primes $5, 19$
Class number $13$
Class group $[13]$
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![2401, -2058, 2107, -1757, 1513, -351, 266, -62, 41, -6, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401)
 
gp: K = bnfinit(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 7 x^{10} - 6 x^{9} + 41 x^{8} - 62 x^{7} + 266 x^{6} - 351 x^{5} + 1513 x^{4} - 1757 x^{3} + 2107 x^{2} - 2058 x + 2401 \)

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:  \(33171021564453125=5^{9}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.81$
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:  $5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Gal(K/\Q)|$:  $12$
This field is Galois and abelian over $\Q$.
Conductor:  \(95=5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{95}(64,·)$, $\chi_{95}(1,·)$, $\chi_{95}(68,·)$, $\chi_{95}(39,·)$, $\chi_{95}(11,·)$, $\chi_{95}(77,·)$, $\chi_{95}(49,·)$, $\chi_{95}(83,·)$, $\chi_{95}(87,·)$, $\chi_{95}(7,·)$, $\chi_{95}(26,·)$, $\chi_{95}(58,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{360437} a^{9} + \frac{60066}{360437} a^{8} + \frac{211}{51491} a^{7} - \frac{1511}{360437} a^{6} + \frac{70398}{360437} a^{5} - \frac{60325}{360437} a^{4} - \frac{8540}{51491} a^{3} - \frac{20595}{360437} a^{2} - \frac{39486}{360437} a - \frac{20301}{51491}$, $\frac{1}{2523059} a^{10} - \frac{1}{2523059} a^{9} - \frac{52716}{360437} a^{8} + \frac{307469}{2523059} a^{7} + \frac{1511}{2523059} a^{6} - \frac{10107}{2523059} a^{5} - \frac{50229}{360437} a^{4} + \frac{111271}{2523059} a^{3} + \frac{1101906}{2523059} a^{2} - \frac{53206}{360437} a + \frac{8828}{51491}$, $\frac{1}{17661413} a^{11} - \frac{1}{17661413} a^{10} + \frac{1}{2523059} a^{9} - \frac{6839769}{17661413} a^{8} + \frac{2584889}{17661413} a^{7} - \frac{1777}{17661413} a^{6} + \frac{422222}{2523059} a^{5} - \frac{2533406}{17661413} a^{4} + \frac{2773982}{17661413} a^{3} + \frac{236860}{2523059} a^{2} - \frac{51204}{360437} a - \frac{10052}{51491}$

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

Class group and class number

$C_{13}$, which has order $13$

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:  \( -\frac{41}{360437} a^{11} - \frac{33}{32767} a^{6} + \frac{294302}{360437} a \) (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:  \( 1234.55163261 \)
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

\(\Q(\sqrt{5}) \), 3.3.361.1, \(\Q(\zeta_{5})\), 6.6.16290125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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}$$[\ ]_{4}^{3}$
$19$19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.5.4t1.a.b$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.19.3t1.a.a$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
* 1.95.12t1.a.a$1$ $ 5 \cdot 19 $ $x^{12} - x^{11} + 7 x^{10} - 6 x^{9} + 41 x^{8} - 62 x^{7} + 266 x^{6} - 351 x^{5} + 1513 x^{4} - 1757 x^{3} + 2107 x^{2} - 2058 x + 2401$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.95.6t1.a.a$1$ $ 5 \cdot 19 $ $x^{6} - x^{5} - 16 x^{4} + x^{3} + 47 x^{2} + 10 x - 11$ $C_6$ (as 6T1) $0$ $1$
* 1.95.12t1.a.b$1$ $ 5 \cdot 19 $ $x^{12} - x^{11} + 7 x^{10} - 6 x^{9} + 41 x^{8} - 62 x^{7} + 266 x^{6} - 351 x^{5} + 1513 x^{4} - 1757 x^{3} + 2107 x^{2} - 2058 x + 2401$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.19.3t1.a.b$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
* 1.95.12t1.a.c$1$ $ 5 \cdot 19 $ $x^{12} - x^{11} + 7 x^{10} - 6 x^{9} + 41 x^{8} - 62 x^{7} + 266 x^{6} - 351 x^{5} + 1513 x^{4} - 1757 x^{3} + 2107 x^{2} - 2058 x + 2401$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.95.6t1.a.b$1$ $ 5 \cdot 19 $ $x^{6} - x^{5} - 16 x^{4} + x^{3} + 47 x^{2} + 10 x - 11$ $C_6$ (as 6T1) $0$ $1$
* 1.95.12t1.a.d$1$ $ 5 \cdot 19 $ $x^{12} - x^{11} + 7 x^{10} - 6 x^{9} + 41 x^{8} - 62 x^{7} + 266 x^{6} - 351 x^{5} + 1513 x^{4} - 1757 x^{3} + 2107 x^{2} - 2058 x + 2401$ $C_{12}$ (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 *.