↑

Properties

Label 12.0.19387378809...3921.2
Degree $12$
Signature $[0, 6]$
Discriminant $3^{18}\cdot 7^{10}\cdot 11^{6}$
Root discriminant $87.22$
Ramified primes $3, 7, 11$
Class number $10368$ (GRH)
Class group $[6, 24, 72]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![157375, -14205, 41817, -10089, 10530, 1203, 832, -600, 204, 72, -9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 9*x^10 + 72*x^9 + 204*x^8 - 600*x^7 + 832*x^6 + 1203*x^5 + 10530*x^4 - 10089*x^3 + 41817*x^2 - 14205*x + 157375)
 
gp: K = bnfinit(x^12 - 6*x^11 - 9*x^10 + 72*x^9 + 204*x^8 - 600*x^7 + 832*x^6 + 1203*x^5 + 10530*x^4 - 10089*x^3 + 41817*x^2 - 14205*x + 157375, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} - 9 x^{10} + 72 x^{9} + 204 x^{8} - 600 x^{7} + 832 x^{6} + 1203 x^{5} + 10530 x^{4} - 10089 x^{3} + 41817 x^{2} - 14205 x + 157375 \)

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:  \(193873788090710808693921=3^{18}\cdot 7^{10}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.22$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(693=3^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{693}(1,·)$, $\chi_{693}(67,·)$, $\chi_{693}(551,·)$, $\chi_{693}(362,·)$, $\chi_{693}(331,·)$, $\chi_{693}(142,·)$, $\chi_{693}(626,·)$, $\chi_{693}(692,·)$, $\chi_{693}(505,·)$, $\chi_{693}(122,·)$, $\chi_{693}(571,·)$, $\chi_{693}(188,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{25} a^{9} - \frac{2}{25} a^{8} - \frac{1}{25} a^{7} + \frac{2}{25} a^{6} - \frac{1}{25} a^{5} + \frac{7}{25} a^{4} - \frac{9}{25} a^{3} - \frac{7}{25} a^{2} + \frac{2}{5} a$, $\frac{1}{125} a^{10} - \frac{1}{125} a^{9} + \frac{12}{125} a^{8} + \frac{6}{125} a^{7} - \frac{4}{125} a^{6} + \frac{11}{125} a^{5} - \frac{42}{125} a^{4} + \frac{4}{125} a^{3} + \frac{8}{125} a^{2} - \frac{9}{25} a$, $\frac{1}{75547495405873875625} a^{11} - \frac{120082531405999842}{75547495405873875625} a^{10} - \frac{492920724491072592}{75547495405873875625} a^{9} - \frac{7553527491817566846}{75547495405873875625} a^{8} + \frac{1170117513920326159}{15109499081174775125} a^{7} + \frac{1027905581672167852}{15109499081174775125} a^{6} - \frac{5369701878705306448}{75547495405873875625} a^{5} + \frac{2741427320755639486}{75547495405873875625} a^{4} - \frac{35319164375719108251}{75547495405873875625} a^{3} + \frac{22681078810096609942}{75547495405873875625} a^{2} - \frac{4494581469636223451}{15109499081174775125} a - \frac{278738678808144516}{604379963246991005}$

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

Class group and class number

$C_{6}\times C_{24}\times C_{72}$, which has order $10368$ (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:  \( 5471.6045925579565 \) (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

\(\Q(\sqrt{-231}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-11}) \), 3.3.3969.2, \(\Q(\sqrt{-11}, \sqrt{21})\), 6.0.440311012911.4, 6.6.330812181.1, 6.0.20967191091.5

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.1.0.1}{1} }^{12}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$3$3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$