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Properties

Label 14.14.4017749625...5952.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{21}\cdot 7^{24}$
Root discriminant $79.48$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group Trivial (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![881, 3500, -3164, -22190, -5292, 37282, 22169, -11184, -7987, 1190, 1043, -42, -56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 56*x^12 - 42*x^11 + 1043*x^10 + 1190*x^9 - 7987*x^8 - 11184*x^7 + 22169*x^6 + 37282*x^5 - 5292*x^4 - 22190*x^3 - 3164*x^2 + 3500*x + 881)
 
gp: K = bnfinit(x^14 - 56*x^12 - 42*x^11 + 1043*x^10 + 1190*x^9 - 7987*x^8 - 11184*x^7 + 22169*x^6 + 37282*x^5 - 5292*x^4 - 22190*x^3 - 3164*x^2 + 3500*x + 881, 1)
 

Normalized defining polynomial

\( x^{14} - 56 x^{12} - 42 x^{11} + 1043 x^{10} + 1190 x^{9} - 7987 x^{8} - 11184 x^{7} + 22169 x^{6} + 37282 x^{5} - 5292 x^{4} - 22190 x^{3} - 3164 x^{2} + 3500 x + 881 \)

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:  \(401774962552217617093885952=2^{21}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.48$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(392=2^{3}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{392}(1,·)$, $\chi_{392}(197,·)$, $\chi_{392}(225,·)$, $\chi_{392}(169,·)$, $\chi_{392}(29,·)$, $\chi_{392}(141,·)$, $\chi_{392}(337,·)$, $\chi_{392}(365,·)$, $\chi_{392}(113,·)$, $\chi_{392}(85,·)$, $\chi_{392}(281,·)$, $\chi_{392}(57,·)$, $\chi_{392}(253,·)$, $\chi_{392}(309,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{31} a^{10} - \frac{14}{31} a^{9} + \frac{7}{31} a^{8} - \frac{13}{31} a^{7} - \frac{2}{31} a^{6} - \frac{11}{31} a^{5} - \frac{6}{31} a^{4} + \frac{10}{31} a^{3} + \frac{6}{31} a^{2} - \frac{3}{31} a - \frac{6}{31}$, $\frac{1}{31} a^{11} - \frac{3}{31} a^{9} - \frac{8}{31} a^{8} + \frac{2}{31} a^{7} - \frac{8}{31} a^{6} - \frac{5}{31} a^{5} - \frac{12}{31} a^{4} - \frac{9}{31} a^{3} - \frac{12}{31} a^{2} + \frac{14}{31} a + \frac{9}{31}$, $\frac{1}{589} a^{12} + \frac{3}{589} a^{11} + \frac{5}{589} a^{10} + \frac{243}{589} a^{9} + \frac{189}{589} a^{8} - \frac{199}{589} a^{7} - \frac{200}{589} a^{6} + \frac{288}{589} a^{5} + \frac{3}{19} a^{4} - \frac{207}{589} a^{3} - \frac{222}{589} a^{2} - \frac{66}{589} a - \frac{238}{589}$, $\frac{1}{108103235692815619} a^{13} + \frac{16726229385387}{108103235692815619} a^{12} - \frac{714636006765791}{108103235692815619} a^{11} + \frac{3658673466200}{108103235692815619} a^{10} - \frac{17309064199381876}{108103235692815619} a^{9} + \frac{37180538527012715}{108103235692815619} a^{8} + \frac{53233452441181129}{108103235692815619} a^{7} - \frac{856164377068916}{108103235692815619} a^{6} + \frac{51613309230747566}{108103235692815619} a^{5} + \frac{33261544072052112}{108103235692815619} a^{4} - \frac{16033303359860959}{108103235692815619} a^{3} - \frac{45061432883031384}{108103235692815619} a^{2} - \frac{17326636119746083}{108103235692815619} a - \frac{24176245364135575}{108103235692815619}$

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:  \( 741991346.6880023 \) (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

\(\Q(\sqrt{2}) \), 7.7.13841287201.1

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 R ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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}$$[3]^{7}$
$7$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$