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Properties

Label 12.0.40712520501...8944.5
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 13^{6}$
Root discriminant $63.21$
Ramified primes $2, 3, 7, 13$
Class number $2688$ (GRH)
Class group $[2, 4, 4, 84]$ (GRH)
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![11827921, -1692824, 3655953, -400386, 511259, -44478, 42260, -2860, 2172, -106, 67, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 67*x^10 - 106*x^9 + 2172*x^8 - 2860*x^7 + 42260*x^6 - 44478*x^5 + 511259*x^4 - 400386*x^3 + 3655953*x^2 - 1692824*x + 11827921)
 
gp: K = bnfinit(x^12 - 2*x^11 + 67*x^10 - 106*x^9 + 2172*x^8 - 2860*x^7 + 42260*x^6 - 44478*x^5 + 511259*x^4 - 400386*x^3 + 3655953*x^2 - 1692824*x + 11827921, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 67 x^{10} - 106 x^{9} + 2172 x^{8} - 2860 x^{7} + 42260 x^{6} - 44478 x^{5} + 511259 x^{4} - 400386 x^{3} + 3655953 x^{2} - 1692824 x + 11827921 \)

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:  \(4071252050148030418944=2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.21$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1092=2^{2}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1092}(1,·)$, $\chi_{1092}(1091,·)$, $\chi_{1092}(677,·)$, $\chi_{1092}(209,·)$, $\chi_{1092}(883,·)$, $\chi_{1092}(521,·)$, $\chi_{1092}(781,·)$, $\chi_{1092}(625,·)$, $\chi_{1092}(467,·)$, $\chi_{1092}(311,·)$, $\chi_{1092}(571,·)$, $\chi_{1092}(415,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{214129537976285608003999657} a^{11} + \frac{82768553954150627811435471}{214129537976285608003999657} a^{10} + \frac{79689712775221265659477215}{214129537976285608003999657} a^{9} + \frac{58758063306386022249811411}{214129537976285608003999657} a^{8} + \frac{67556628306818230399793579}{214129537976285608003999657} a^{7} - \frac{104789915094663375439376116}{214129537976285608003999657} a^{6} + \frac{7863221960967095742366472}{214129537976285608003999657} a^{5} + \frac{53067690337507915722729319}{214129537976285608003999657} a^{4} - \frac{58094841667311769620323784}{214129537976285608003999657} a^{3} + \frac{27844418817461188162740774}{214129537976285608003999657} a^{2} - \frac{51026627368583265753551155}{214129537976285608003999657} a + \frac{44523422610587381561554974}{214129537976285608003999657}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{84}$, which has order $2688$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 140.7987960054707 \) (assuming GRH)
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

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-273}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-13}, \sqrt{21})\), \(\Q(\zeta_{21})^+\), 6.0.337599808.1, 6.0.63806363712.2

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$