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Properties

Label 14.0.12111141100...8711.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 1277^{7}$
Root discriminant $61.90$
Ramified primes $3, 1277$
Class number $1014$ (GRH)
Class group $[13, 78]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28167, -72588, 116395, -127299, 153774, -49512, 13383, 9867, 1873, 237, -567, -60, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 12*x^12 - 60*x^11 - 567*x^10 + 237*x^9 + 1873*x^8 + 9867*x^7 + 13383*x^6 - 49512*x^5 + 153774*x^4 - 127299*x^3 + 116395*x^2 - 72588*x + 28167)
 
gp: K = bnfinit(x^14 + 12*x^12 - 60*x^11 - 567*x^10 + 237*x^9 + 1873*x^8 + 9867*x^7 + 13383*x^6 - 49512*x^5 + 153774*x^4 - 127299*x^3 + 116395*x^2 - 72588*x + 28167, 1)
 

Normalized defining polynomial

\( x^{14} + 12 x^{12} - 60 x^{11} - 567 x^{10} + 237 x^{9} + 1873 x^{8} + 9867 x^{7} + 13383 x^{6} - 49512 x^{5} + 153774 x^{4} - 127299 x^{3} + 116395 x^{2} - 72588 x + 28167 \)

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:  \(-12111141100130785779768711=-\,3^{7}\cdot 1277^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.90$
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:  $3, 1277$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{10} - \frac{1}{27} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{8}{27} a^{4} + \frac{1}{3} a^{3} - \frac{8}{27} a^{2} - \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{9} + \frac{1}{9} a^{7} - \frac{1}{27} a^{5} - \frac{1}{3} a^{4} + \frac{10}{27} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{207009} a^{12} - \frac{2164}{207009} a^{11} - \frac{676}{69003} a^{10} + \frac{73}{207009} a^{9} + \frac{6605}{207009} a^{8} - \frac{79}{2091} a^{7} + \frac{22835}{207009} a^{6} + \frac{85522}{207009} a^{5} + \frac{743}{1683} a^{4} + \frac{5197}{18819} a^{3} + \frac{90136}{207009} a^{2} + \frac{3577}{7667} a + \frac{256}{1683}$, $\frac{1}{51996465005524233708059469} a^{13} + \frac{123983711832701323762}{51996465005524233708059469} a^{12} - \frac{17702033512614263899720}{3058615588560249041650557} a^{11} - \frac{479312065722950274927806}{51996465005524233708059469} a^{10} + \frac{1639755846411914540554198}{51996465005524233708059469} a^{9} - \frac{2114189310237406212057413}{51996465005524233708059469} a^{8} - \frac{441277890691221180151156}{7428066429360604815437067} a^{7} - \frac{8097505813754381077492015}{51996465005524233708059469} a^{6} - \frac{3216901460806014705667960}{51996465005524233708059469} a^{5} - \frac{2322431030577519220426813}{51996465005524233708059469} a^{4} + \frac{258172454887764187753919}{51996465005524233708059469} a^{3} + \frac{1136215942126674774419632}{3058615588560249041650557} a^{2} - \frac{2096081564402084251683928}{17332155001841411236019823} a + \frac{138405497478560515995644}{422735487849790517951703}$

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

Class group and class number

$C_{13}\times C_{78}$, which has order $1014$ (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:  \( 70433.5060458 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-3831}) \), 7.1.56225905191.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.56225905191.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
1277Data not computed