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Properties

Label 14.0.31178282477...9552.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 3^{8}\cdot 7^{20}\cdot 73^{4}$
Root discriminant $290.97$
Ramified primes $2, 3, 7, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T26

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1476661248, 540954624, 397307904, 117175296, 40943168, 5717824, 1564080, 35808, -2212, -16044, -623, -364, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 42*x^12 - 364*x^11 - 623*x^10 - 16044*x^9 - 2212*x^8 + 35808*x^7 + 1564080*x^6 + 5717824*x^5 + 40943168*x^4 + 117175296*x^3 + 397307904*x^2 + 540954624*x + 1476661248)
 
gp: K = bnfinit(x^14 + 42*x^12 - 364*x^11 - 623*x^10 - 16044*x^9 - 2212*x^8 + 35808*x^7 + 1564080*x^6 + 5717824*x^5 + 40943168*x^4 + 117175296*x^3 + 397307904*x^2 + 540954624*x + 1476661248, 1)
 

Normalized defining polynomial

\( x^{14} + 42 x^{12} - 364 x^{11} - 623 x^{10} - 16044 x^{9} - 2212 x^{8} + 35808 x^{7} + 1564080 x^{6} + 5717824 x^{5} + 40943168 x^{4} + 117175296 x^{3} + 397307904 x^{2} + 540954624 x + 1476661248 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-31178282477375587773786611936919552=-\,2^{21}\cdot 3^{8}\cdot 7^{20}\cdot 73^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $290.97$
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:  $2, 3, 7, 73$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{7} - \frac{1}{32} a^{6} - \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{64} a^{8} - \frac{1}{64} a^{7} - \frac{1}{64} a^{6} + \frac{1}{64} a^{5} - \frac{1}{16} a^{4} - \frac{1}{16} a^{3}$, $\frac{1}{768} a^{9} - \frac{1}{128} a^{8} - \frac{5}{384} a^{7} + \frac{5}{192} a^{6} - \frac{11}{768} a^{5} + \frac{17}{384} a^{4} + \frac{23}{192} a^{3} + \frac{5}{32} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{3072} a^{10} + \frac{7}{1536} a^{8} - \frac{7}{768} a^{7} - \frac{71}{3072} a^{6} + \frac{13}{768} a^{5} + \frac{17}{192} a^{4} - \frac{1}{64} a^{3} - \frac{5}{64} a^{2} + \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{12288} a^{11} - \frac{1}{6144} a^{10} - \frac{1}{6144} a^{9} - \frac{1}{1536} a^{8} - \frac{95}{12288} a^{7} + \frac{3}{2048} a^{6} - \frac{71}{1536} a^{5} + \frac{7}{768} a^{4} - \frac{17}{768} a^{3} + \frac{1}{128} a^{2} - \frac{7}{32} a + \frac{1}{16}$, $\frac{1}{2727936} a^{12} + \frac{13}{454656} a^{11} + \frac{131}{1363968} a^{10} - \frac{53}{340992} a^{9} - \frac{3631}{2727936} a^{8} - \frac{4031}{1363968} a^{7} + \frac{613}{170496} a^{6} - \frac{2215}{170496} a^{5} + \frac{11335}{170496} a^{4} + \frac{1637}{28416} a^{3} + \frac{209}{1776} a^{2} + \frac{1349}{3552} a - \frac{147}{296}$, $\frac{1}{200913068275054580845934641152} a^{13} + \frac{609364796310854010539}{25114133534381822605741830144} a^{12} - \frac{1377371444437068303370735}{100456534137527290422967320576} a^{11} - \frac{1111810173748422325502069}{16742755689587881737161220096} a^{10} - \frac{1765218471908812713405299}{5430082926352826509349584896} a^{9} - \frac{374239568319792665400827549}{50228267068763645211483660288} a^{8} - \frac{73901809582813045260245531}{5580918563195960579053740032} a^{7} + \frac{1127981271229580729540563}{110149708484130800902376448} a^{6} - \frac{4735756300385342206133959}{220299416968261601804752896} a^{5} - \frac{388551235862727592831805621}{3139266691797727825717728768} a^{4} + \frac{35250205764169900128625175}{348807410199747536190858752} a^{3} + \frac{121650170496509983515165}{1362528946092763813245542} a^{2} + \frac{43610369589028150844568595}{130802778824905326071572032} a + \frac{2079870521362592243521591}{5450115784371055252982168}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $6$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 177351190608000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T26:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 882
The 20 conjugacy class representatives for 1/2[1/2.F_42(7)^2]2
Character table for 1/2[1/2.F_42(7)^2]2

Intermediate fields

\(\Q(\sqrt{-2}) \)

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

Sibling fields

Degree 21 siblings: data not computed
Degree 42 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
7Data not computed
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$
73.7.0.1$x^{7} - 3 x + 13$$1$$7$$0$$C_7$$[\ ]^{7}$