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Properties

Label 12.0.36138776487264256.6
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 13^{10}$
Root discriminant $23.98$
Ramified primes $2, 13$
Class number $9$
Class group $[9]$
Galois group $C_6\times S_3$ (as 12T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 132, 130, 36, 347, -74, 202, -30, 61, -10, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 13*x^10 - 10*x^9 + 61*x^8 - 30*x^7 + 202*x^6 - 74*x^5 + 347*x^4 + 36*x^3 + 130*x^2 + 132*x + 27)
 
gp: K = bnfinit(x^12 - 2*x^11 + 13*x^10 - 10*x^9 + 61*x^8 - 30*x^7 + 202*x^6 - 74*x^5 + 347*x^4 + 36*x^3 + 130*x^2 + 132*x + 27, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 13 x^{10} - 10 x^{9} + 61 x^{8} - 30 x^{7} + 202 x^{6} - 74 x^{5} + 347 x^{4} + 36 x^{3} + 130 x^{2} + 132 x + 27 \)

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:  \(36138776487264256=2^{18}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.98$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} + \frac{4}{9} a^{6} - \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{1409782563} a^{11} + \frac{6885350}{156642507} a^{10} - \frac{3476930}{1409782563} a^{9} + \frac{22080004}{1409782563} a^{8} + \frac{26965756}{67132503} a^{7} + \frac{139082206}{469927521} a^{6} - \frac{118748717}{1409782563} a^{5} - \frac{4130002}{22377501} a^{4} - \frac{435713953}{1409782563} a^{3} - \frac{677253446}{1409782563} a^{2} + \frac{57725825}{469927521} a - \frac{17084432}{52214169}$

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

Class group and class number

$C_{9}$, 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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1689.30454199 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 12T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $C_6\times S_3$
Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-2}, \sqrt{13})\), 6.0.190102016.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.13.2t1.1c1$1$ $ 13 $ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
* 1.2e3_13.2t1.2c1$1$ $ 2^{3} \cdot 13 $ $x^{2} + 26$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e3.2t1.2c1$1$ $ 2^{3}$ $x^{2} + 2$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_13.6t1.3c1$1$ $ 2^{3} \cdot 13 $ $x^{6} - 2 x^{5} - x^{4} - 2 x^{3} + 34 x^{2} + 28 x + 73$ $C_6$ (as 6T1) $0$ $-1$
1.13.6t1.1c1$1$ $ 13 $ $x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
1.13.6t1.1c2$1$ $ 13 $ $x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
1.2e3_13.6t1.4c1$1$ $ 2^{3} \cdot 13 $ $x^{6} + 26 x^{4} + 104 x^{2} + 104$ $C_6$ (as 6T1) $0$ $-1$
1.2e3_13.6t1.4c2$1$ $ 2^{3} \cdot 13 $ $x^{6} + 26 x^{4} + 104 x^{2} + 104$ $C_6$ (as 6T1) $0$ $-1$
1.2e3_13.6t1.3c2$1$ $ 2^{3} \cdot 13 $ $x^{6} - 2 x^{5} - x^{4} - 2 x^{3} + 34 x^{2} + 28 x + 73$ $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
2.2e3_13.3t2.1c1$2$ $ 2^{3} \cdot 13 $ $x^{3} - x - 2$ $S_3$ (as 3T2) $1$ $0$
2.2e3_13.6t3.2c1$2$ $ 2^{3} \cdot 13 $ $x^{6} - x^{5} + 2 x^{4} + x^{3} - 2 x^{2} - x - 1$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e3_13e2.6t5.1c1$2$ $ 2^{3} \cdot 13^{2}$ $x^{6} - 2 x^{5} - 7 x^{4} + 6 x^{3} + 44 x^{2} + 60 x + 27$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e3_13e2.12t18.2c1$2$ $ 2^{3} \cdot 13^{2}$ $x^{12} - 2 x^{11} + 13 x^{10} - 10 x^{9} + 61 x^{8} - 30 x^{7} + 202 x^{6} - 74 x^{5} + 347 x^{4} + 36 x^{3} + 130 x^{2} + 132 x + 27$ $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.2e3_13e2.6t5.1c2$2$ $ 2^{3} \cdot 13^{2}$ $x^{6} - 2 x^{5} - 7 x^{4} + 6 x^{3} + 44 x^{2} + 60 x + 27$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e3_13e2.12t18.2c2$2$ $ 2^{3} \cdot 13^{2}$ $x^{12} - 2 x^{11} + 13 x^{10} - 10 x^{9} + 61 x^{8} - 30 x^{7} + 202 x^{6} - 74 x^{5} + 347 x^{4} + 36 x^{3} + 130 x^{2} + 132 x + 27$ $C_6\times S_3$ (as 12T18) $0$ $0$

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 *.