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Properties

Label 14.14.7738998952...8528.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{21}\cdot 577^{6}$
Root discriminant $43.14$
Ramified primes $2, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{14}$ (as 14T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25, -310, 1, 1886, 60, -3370, 275, 2486, -458, -772, 183, 96, -24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 24*x^12 + 96*x^11 + 183*x^10 - 772*x^9 - 458*x^8 + 2486*x^7 + 275*x^6 - 3370*x^5 + 60*x^4 + 1886*x^3 + x^2 - 310*x - 25)
 
gp: K = bnfinit(x^14 - 4*x^13 - 24*x^12 + 96*x^11 + 183*x^10 - 772*x^9 - 458*x^8 + 2486*x^7 + 275*x^6 - 3370*x^5 + 60*x^4 + 1886*x^3 + x^2 - 310*x - 25, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 24 x^{12} + 96 x^{11} + 183 x^{10} - 772 x^{9} - 458 x^{8} + 2486 x^{7} + 275 x^{6} - 3370 x^{5} + 60 x^{4} + 1886 x^{3} + x^{2} - 310 x - 25 \)

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:  \(77389989525273630998528=2^{21}\cdot 577^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.14$
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:  $2, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{205} a^{12} - \frac{7}{205} a^{11} - \frac{48}{205} a^{10} - \frac{12}{41} a^{9} + \frac{63}{205} a^{8} + \frac{99}{205} a^{7} + \frac{20}{41} a^{6} - \frac{14}{205} a^{5} - \frac{83}{205} a^{4} - \frac{31}{205} a^{3} - \frac{7}{205} a^{2} + \frac{22}{205} a + \frac{9}{41}$, $\frac{1}{407705667265} a^{13} + \frac{413284424}{407705667265} a^{12} + \frac{3410963648}{407705667265} a^{11} + \frac{35522029689}{407705667265} a^{10} + \frac{170474119356}{407705667265} a^{9} - \frac{4193315229}{81541133453} a^{8} + \frac{4513039042}{31361974405} a^{7} - \frac{2677775141}{81541133453} a^{6} - \frac{75767559703}{407705667265} a^{5} - \frac{127365144902}{407705667265} a^{4} - \frac{2651365265}{81541133453} a^{3} - \frac{2969779617}{81541133453} a^{2} - \frac{200412050484}{407705667265} a - \frac{2995058296}{81541133453}$

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:  \( 9330895.34512 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{14}$ (as 14T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 28
The 10 conjugacy class representatives for $D_{14}$
Character table for $D_{14}$

Intermediate fields

\(\Q(\sqrt{2}) \), 7.7.192100033.1

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 14 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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$2.14.21.34$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$2$$7$$21$$C_{14}$$[3]^{7}$
577Data not computed