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Properties

Label 12.0.1593224064453125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 13^{8}$
Root discriminant $18.49$
Ramified primes $5, 13$
Class number $4$
Class group $[2, 2]$
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![1, 4, 15, 55, 201, 85, 84, 50, 31, -10, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 5*x^10 - 10*x^9 + 31*x^8 + 50*x^7 + 84*x^6 + 85*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^12 - x^11 + 5*x^10 - 10*x^9 + 31*x^8 + 50*x^7 + 84*x^6 + 85*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 31 x^{8} + 50 x^{7} + 84 x^{6} + 85 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1 \)

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:  \(1593224064453125=5^{9}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.49$
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:  $5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(1,·)$, $\chi_{65}(3,·)$, $\chi_{65}(16,·)$, $\chi_{65}(9,·)$, $\chi_{65}(42,·)$, $\chi_{65}(14,·)$, $\chi_{65}(29,·)$, $\chi_{65}(48,·)$, $\chi_{65}(53,·)$, $\chi_{65}(22,·)$, $\chi_{65}(27,·)$, $\chi_{65}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6781} a^{9} + \frac{675}{6781} a^{8} + \frac{1298}{6781} a^{7} + \frac{1401}{6781} a^{6} + \frac{3116}{6781} a^{5} + \frac{658}{6781} a^{4} + \frac{3385}{6781} a^{3} - \frac{322}{6781} a^{2} - \frac{358}{6781} a + \frac{2466}{6781}$, $\frac{1}{6781} a^{10} - \frac{532}{6781} a^{5} - \frac{3205}{6781}$, $\frac{1}{6781} a^{11} - \frac{532}{6781} a^{6} - \frac{3205}{6781} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{134}{6781} a^{11} - \frac{670}{6781} a^{10} + \frac{1340}{6781} a^{9} - \frac{4154}{6781} a^{8} + \frac{10165}{6781} a^{7} - \frac{11256}{6781} a^{6} - \frac{11390}{6781} a^{5} - \frac{26934}{6781} a^{4} - \frac{7370}{6781} a^{3} - \frac{92430}{6781} a^{2} - \frac{536}{6781} a - \frac{134}{6781} \) (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:  \( 615.54450504 \)
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

\(\Q(\sqrt{5}) \), 3.3.169.1, \(\Q(\zeta_{5})\), 6.6.3570125.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$13$13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$

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) $0$ $-1$
* 1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.5_13.12t1.1c1$1$ $ 5 \cdot 13 $ $x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 31 x^{8} + 50 x^{7} + 84 x^{6} + 85 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_13.6t1.1c1$1$ $ 5 \cdot 13 $ $x^{6} - x^{5} - 12 x^{4} + 13 x^{3} + 19 x^{2} - 10 x - 5$ $C_6$ (as 6T1) $0$ $1$
* 1.5_13.12t1.1c2$1$ $ 5 \cdot 13 $ $x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 31 x^{8} + 50 x^{7} + 84 x^{6} + 85 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.5_13.12t1.1c3$1$ $ 5 \cdot 13 $ $x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 31 x^{8} + 50 x^{7} + 84 x^{6} + 85 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_13.6t1.1c2$1$ $ 5 \cdot 13 $ $x^{6} - x^{5} - 12 x^{4} + 13 x^{3} + 19 x^{2} - 10 x - 5$ $C_6$ (as 6T1) $0$ $1$
* 1.5_13.12t1.1c4$1$ $ 5 \cdot 13 $ $x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 31 x^{8} + 50 x^{7} + 84 x^{6} + 85 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1$ $C_{12}$ (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 *.