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Properties

Label 14.14.8238029474...0625.1
Degree $14$
Signature $[14, 0]$
Discriminant $5^{7}\cdot 11^{3}\cdot 311^{3}\cdot 641^{3}$
Root discriminant $51.08$
Ramified primes $5, 11, 311, 641$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_7 \wr C_2$ (as 14T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -70, 373, -299, -2272, 2236, 6281, 332, -2618, -324, 441, 47, -34, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 34*x^12 + 47*x^11 + 441*x^10 - 324*x^9 - 2618*x^8 + 332*x^7 + 6281*x^6 + 2236*x^5 - 2272*x^4 - 299*x^3 + 373*x^2 - 70*x + 4)
 
gp: K = bnfinit(x^14 - 2*x^13 - 34*x^12 + 47*x^11 + 441*x^10 - 324*x^9 - 2618*x^8 + 332*x^7 + 6281*x^6 + 2236*x^5 - 2272*x^4 - 299*x^3 + 373*x^2 - 70*x + 4, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 34 x^{12} + 47 x^{11} + 441 x^{10} - 324 x^{9} - 2618 x^{8} + 332 x^{7} + 6281 x^{6} + 2236 x^{5} - 2272 x^{4} - 299 x^{3} + 373 x^{2} - 70 x + 4 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(823802947443529170390625=5^{7}\cdot 11^{3}\cdot 311^{3}\cdot 641^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.08$
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:  $5, 11, 311, 641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{4} - \frac{7}{16} a^{3} - \frac{1}{16} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{11} + \frac{1}{64} a^{10} - \frac{1}{32} a^{9} + \frac{7}{64} a^{8} - \frac{11}{64} a^{7} + \frac{1}{16} a^{6} + \frac{13}{32} a^{5} - \frac{29}{64} a^{4} + \frac{1}{8} a^{3} - \frac{27}{64} a^{2} + \frac{13}{32} a - \frac{1}{16}$, $\frac{1}{256} a^{12} - \frac{3}{256} a^{10} + \frac{9}{256} a^{9} + \frac{7}{128} a^{8} + \frac{47}{256} a^{7} - \frac{5}{128} a^{6} - \frac{119}{256} a^{5} + \frac{101}{256} a^{4} - \frac{35}{256} a^{3} + \frac{53}{256} a^{2} - \frac{31}{128} a + \frac{1}{64}$, $\frac{1}{1024} a^{13} - \frac{1}{1024} a^{12} - \frac{3}{1024} a^{11} - \frac{5}{256} a^{10} + \frac{5}{1024} a^{9} + \frac{97}{1024} a^{8} + \frac{39}{1024} a^{7} - \frac{237}{1024} a^{6} + \frac{87}{256} a^{5} - \frac{41}{128} a^{4} + \frac{55}{128} a^{3} - \frac{211}{1024} a^{2} + \frac{129}{512} a - \frac{97}{256}$

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:  $13$
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:  \( 45374712.1437 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7\wr C_2$ (as 14T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 392
The 20 conjugacy class representatives for $D_7 \wr C_2$
Character table for $D_7 \wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \)

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

Sibling fields

Degree 14 sibling: data not computed
Degree 28 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.7.0.1$x^{7} - x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
311Data not computed
641Data not computed