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Properties

Label 12.4.14116592251...0832.9
Degree $12$
Signature $[4, 4]$
Discriminant $2^{25}\cdot 29^{10}$
Root discriminant $70.11$
Ramified primes $2, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $A_4:C_4$ (as 12T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14380, 15209, -75901, 20679, -2703, 5030, 1042, 1050, 194, 17, 11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 11*x^10 + 17*x^9 + 194*x^8 + 1050*x^7 + 1042*x^6 + 5030*x^5 - 2703*x^4 + 20679*x^3 - 75901*x^2 + 15209*x + 14380)
 
gp: K = bnfinit(x^12 - x^11 + 11*x^10 + 17*x^9 + 194*x^8 + 1050*x^7 + 1042*x^6 + 5030*x^5 - 2703*x^4 + 20679*x^3 - 75901*x^2 + 15209*x + 14380, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 11 x^{10} + 17 x^{9} + 194 x^{8} + 1050 x^{7} + 1042 x^{6} + 5030 x^{5} - 2703 x^{4} + 20679 x^{3} - 75901 x^{2} + 15209 x + 14380 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14116592251679730040832=2^{25}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.11$
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:  $2, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{3}{8} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{5} + \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{3}{16} a - \frac{1}{4}$, $\frac{1}{32} a^{8} - \frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{1}{32} a^{5} - \frac{5}{32} a^{4} - \frac{7}{32} a^{3} + \frac{1}{32} a^{2} - \frac{9}{32} a - \frac{3}{8}$, $\frac{1}{448} a^{9} - \frac{3}{224} a^{8} - \frac{1}{224} a^{6} + \frac{1}{224} a^{5} - \frac{55}{224} a^{4} + \frac{3}{56} a^{3} + \frac{3}{32} a^{2} + \frac{165}{448} a + \frac{3}{112}$, $\frac{1}{13440} a^{10} - \frac{11}{13440} a^{9} - \frac{83}{6720} a^{8} - \frac{43}{6720} a^{7} + \frac{1}{140} a^{6} - \frac{317}{3360} a^{5} + \frac{1}{64} a^{4} + \frac{1}{2240} a^{3} - \frac{1611}{4480} a^{2} - \frac{1531}{4480} a + \frac{193}{672}$, $\frac{1}{2098772264830080} a^{11} + \frac{8758067209}{1049386132415040} a^{10} - \frac{9795400125}{19988307284096} a^{9} + \frac{1238458909}{4997076821024} a^{8} - \frac{11262459172759}{1049386132415040} a^{7} - \frac{12170634364271}{524693066207520} a^{6} - \frac{83513814374041}{1049386132415040} a^{5} + \frac{3248901986509}{87448844367920} a^{4} - \frac{355926013913}{17063189144960} a^{3} + \frac{19396996529059}{69959075494336} a^{2} - \frac{730727581975897}{2098772264830080} a - \frac{4973679147587}{104938613241504}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $7$
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:  \( 56489532.2876 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4:C_4$ (as 12T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $A_4:C_4$
Character table for $A_4:C_4$

Intermediate fields

3.3.6728.1, 6.2.1448511488.7

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

Sibling fields

Degree 12 sibling: data not computed
Degree 16 sibling: data not computed
Degree 24 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
$29$29.12.10.3$x^{12} + 232 x^{6} + 22707$$6$$2$$10$$C_3 : C_4$$[\ ]_{6}^{2}$