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Properties

Label 12.2.338362575386312704.1
Degree $12$
Signature $[2, 5]$
Discriminant $-\,2^{20}\cdot 19^{9}$
Root discriminant $28.89$
Ramified primes $2, 19$
Class number $2$
Class group $[2]$
Galois group $(C_6\times C_2):C_2$ (as 12T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-927, 2082, -1624, -150, 1163, -804, 120, 132, -69, -6, 16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 16*x^10 - 6*x^9 - 69*x^8 + 132*x^7 + 120*x^6 - 804*x^5 + 1163*x^4 - 150*x^3 - 1624*x^2 + 2082*x - 927)
 
gp: K = bnfinit(x^12 - 6*x^11 + 16*x^10 - 6*x^9 - 69*x^8 + 132*x^7 + 120*x^6 - 804*x^5 + 1163*x^4 - 150*x^3 - 1624*x^2 + 2082*x - 927, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 16 x^{10} - 6 x^{9} - 69 x^{8} + 132 x^{7} + 120 x^{6} - 804 x^{5} + 1163 x^{4} - 150 x^{3} - 1624 x^{2} + 2082 x - 927 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-338362575386312704=-\,2^{20}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.89$
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:  $2, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{12} a^{9} - \frac{1}{12} a$, $\frac{1}{552} a^{10} - \frac{5}{276} a^{9} + \frac{13}{184} a^{8} + \frac{9}{92} a^{7} - \frac{5}{23} a^{6} - \frac{5}{92} a^{5} + \frac{3}{23} a^{4} + \frac{41}{92} a^{3} - \frac{217}{552} a^{2} + \frac{31}{138} a - \frac{29}{184}$, $\frac{1}{4502664} a^{11} - \frac{293}{1500888} a^{10} + \frac{180079}{4502664} a^{9} + \frac{28195}{500296} a^{8} - \frac{27253}{750444} a^{7} + \frac{120277}{750444} a^{6} + \frac{174557}{750444} a^{5} - \frac{113519}{750444} a^{4} + \frac{751457}{4502664} a^{3} - \frac{425345}{1500888} a^{2} + \frac{937967}{4502664} a + \frac{453185}{1500888}$

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

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
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:  \( 19015.3365166 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:D_4$ (as 12T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 9 conjugacy class representatives for $(C_6\times C_2):C_2$
Character table for $(C_6\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{19}) \), 3.1.76.1, 4.2.438976.1, 6.2.1755904.2

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 12 sibling: 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.12.20.65$x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{4} - 2$$12$$1$$20$$(C_6\times C_2):C_2$$[2, 2]_{3}^{2}$
$19$19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.8.6.2$x^{8} - 19 x^{4} + 5776$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$