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Properties

Label 14.0.22358793885...3567.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,127^{13}$
Root discriminant $89.85$
Ramified prime $127$
Class number $215$ (GRH)
Class group $[215]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![72493, 187457, 157025, 21289, -21241, 4960, 10394, 5527, 2049, -836, -244, 53, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 5*x^12 + 53*x^11 - 244*x^10 - 836*x^9 + 2049*x^8 + 5527*x^7 + 10394*x^6 + 4960*x^5 - 21241*x^4 + 21289*x^3 + 157025*x^2 + 187457*x + 72493)
 
gp: K = bnfinit(x^14 - x^13 + 5*x^12 + 53*x^11 - 244*x^10 - 836*x^9 + 2049*x^8 + 5527*x^7 + 10394*x^6 + 4960*x^5 - 21241*x^4 + 21289*x^3 + 157025*x^2 + 187457*x + 72493, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 5 x^{12} + 53 x^{11} - 244 x^{10} - 836 x^{9} + 2049 x^{8} + 5527 x^{7} + 10394 x^{6} + 4960 x^{5} - 21241 x^{4} + 21289 x^{3} + 157025 x^{2} + 187457 x + 72493 \)

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:  \(-2235879388560037062539773567=-\,127^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.85$
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:  $127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(127\)
Dirichlet character group:    $\lbrace$$\chi_{127}(32,·)$, $\chi_{127}(1,·)$, $\chi_{127}(2,·)$, $\chi_{127}(4,·)$, $\chi_{127}(8,·)$, $\chi_{127}(64,·)$, $\chi_{127}(63,·)$, $\chi_{127}(111,·)$, $\chi_{127}(16,·)$, $\chi_{127}(119,·)$, $\chi_{127}(123,·)$, $\chi_{127}(125,·)$, $\chi_{127}(126,·)$, $\chi_{127}(95,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9} + \frac{4}{19} a^{8} + \frac{7}{19} a^{7} - \frac{4}{19} a^{5} - \frac{5}{19} a^{4} + \frac{7}{19} a^{3} - \frac{1}{19} a^{2} - \frac{3}{19} a - \frac{9}{19}$, $\frac{1}{19} a^{10} - \frac{9}{19} a^{8} - \frac{9}{19} a^{7} - \frac{4}{19} a^{6} - \frac{8}{19} a^{5} + \frac{8}{19} a^{4} + \frac{9}{19} a^{3} + \frac{1}{19} a^{2} + \frac{3}{19} a - \frac{2}{19}$, $\frac{1}{19} a^{11} + \frac{8}{19} a^{8} + \frac{2}{19} a^{7} - \frac{8}{19} a^{6} - \frac{9}{19} a^{5} + \frac{2}{19} a^{4} + \frac{7}{19} a^{3} - \frac{6}{19} a^{2} + \frac{9}{19} a - \frac{5}{19}$, $\frac{1}{38627} a^{12} + \frac{124}{38627} a^{11} + \frac{160}{38627} a^{10} - \frac{169}{38627} a^{9} - \frac{10540}{38627} a^{8} - \frac{15986}{38627} a^{7} + \frac{3223}{38627} a^{6} + \frac{5192}{38627} a^{5} + \frac{16822}{38627} a^{4} + \frac{11684}{38627} a^{3} - \frac{5718}{38627} a^{2} + \frac{9247}{38627} a + \frac{14276}{38627}$, $\frac{1}{2112227343892961157398293} a^{13} + \frac{10214421585082390363}{2112227343892961157398293} a^{12} - \frac{39759920292003755699066}{2112227343892961157398293} a^{11} + \frac{46289550066330923654844}{2112227343892961157398293} a^{10} + \frac{43110616455054841591221}{2112227343892961157398293} a^{9} + \frac{85396830601528731312728}{2112227343892961157398293} a^{8} + \frac{595393954977151548112157}{2112227343892961157398293} a^{7} + \frac{891041461521983501070239}{2112227343892961157398293} a^{6} + \frac{434741313217475524976481}{2112227343892961157398293} a^{5} - \frac{167578874923921484553416}{2112227343892961157398293} a^{4} + \frac{225168239066282985066926}{2112227343892961157398293} a^{3} + \frac{41343228593633810257160}{111169860204892692494647} a^{2} - \frac{700725873098672271409945}{2112227343892961157398293} a - \frac{763924996540964878626076}{2112227343892961157398293}$

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

Class group and class number

$C_{215}$, which has order $215$ (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:  \( 546287.210347 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

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

Intermediate fields

\(\Q(\sqrt{-127}) \), 7.7.4195872914689.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$127$127.14.13.11$x^{14} + 607437063$$14$$1$$13$$C_{14}$$[\ ]_{14}$