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Properties

Label 12.0.6818265813529681.1
Degree $12$
Signature $[0, 6]$
Discriminant $7^{10}\cdot 17^{6}$
Root discriminant $20.87$
Ramified primes $7, 17$
Class number $5$
Class group $[5]$
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 1024, 1280, 576, 464, 260, 181, -65, 29, -9, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 5*x^10 - 9*x^9 + 29*x^8 - 65*x^7 + 181*x^6 + 260*x^5 + 464*x^4 + 576*x^3 + 1280*x^2 + 1024*x + 4096)
 
gp: K = bnfinit(x^12 - x^11 + 5*x^10 - 9*x^9 + 29*x^8 - 65*x^7 + 181*x^6 + 260*x^5 + 464*x^4 + 576*x^3 + 1280*x^2 + 1024*x + 4096, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 5 x^{10} - 9 x^{9} + 29 x^{8} - 65 x^{7} + 181 x^{6} + 260 x^{5} + 464 x^{4} + 576 x^{3} + 1280 x^{2} + 1024 x + 4096 \)

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:  \(6818265813529681=7^{10}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.87$
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:  $7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(119=7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{119}(1,·)$, $\chi_{119}(67,·)$, $\chi_{119}(69,·)$, $\chi_{119}(103,·)$, $\chi_{119}(18,·)$, $\chi_{119}(16,·)$, $\chi_{119}(50,·)$, $\chi_{119}(52,·)$, $\chi_{119}(86,·)$, $\chi_{119}(33,·)$, $\chi_{119}(118,·)$, $\chi_{119}(101,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{724} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{65}{181}$, $\frac{1}{2896} a^{8} - \frac{1}{2896} a^{7} + \frac{1}{16} a^{6} - \frac{5}{16} a^{5} - \frac{7}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{65}{724} a + \frac{29}{181}$, $\frac{1}{11584} a^{9} - \frac{1}{11584} a^{8} + \frac{5}{11584} a^{7} - \frac{5}{64} a^{6} + \frac{9}{64} a^{5} - \frac{29}{64} a^{4} + \frac{1}{64} a^{3} + \frac{65}{2896} a^{2} + \frac{29}{724} a + \frac{9}{181}$, $\frac{1}{46336} a^{10} - \frac{1}{46336} a^{9} + \frac{5}{46336} a^{8} - \frac{9}{46336} a^{7} - \frac{55}{256} a^{6} + \frac{35}{256} a^{5} + \frac{1}{256} a^{4} + \frac{65}{11584} a^{3} + \frac{29}{2896} a^{2} + \frac{9}{724} a + \frac{5}{181}$, $\frac{1}{185344} a^{11} - \frac{1}{185344} a^{10} + \frac{5}{185344} a^{9} - \frac{9}{185344} a^{8} + \frac{29}{185344} a^{7} - \frac{221}{1024} a^{6} + \frac{1}{1024} a^{5} + \frac{65}{46336} a^{4} + \frac{29}{11584} a^{3} + \frac{9}{2896} a^{2} + \frac{5}{724} a + \frac{1}{181}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( -\frac{1}{2896} a^{9} - \frac{1165}{2896} a^{2} \) (order $14$)
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:  \( 1059.54542703 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-119}) \), \(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-7}, \sqrt{17})\), \(\Q(\zeta_{7})\), 6.0.82572791.1, 6.6.11796113.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$