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Properties

Label 14.0.91716860308...6671.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,19^{7}\cdot 29^{13}$
Root discriminant $99.38$
Ramified primes $19, 29$
Class number $14768$ (GRH)
Class group $[2, 2, 2, 1846]$ (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![14031649, -17492952, 18283513, -6549159, 3501044, -1853951, 809745, -144735, 108150, -6000, 5866, -133, 132, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 132*x^12 - 133*x^11 + 5866*x^10 - 6000*x^9 + 108150*x^8 - 144735*x^7 + 809745*x^6 - 1853951*x^5 + 3501044*x^4 - 6549159*x^3 + 18283513*x^2 - 17492952*x + 14031649)
 
gp: K = bnfinit(x^14 - x^13 + 132*x^12 - 133*x^11 + 5866*x^10 - 6000*x^9 + 108150*x^8 - 144735*x^7 + 809745*x^6 - 1853951*x^5 + 3501044*x^4 - 6549159*x^3 + 18283513*x^2 - 17492952*x + 14031649, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 132 x^{12} - 133 x^{11} + 5866 x^{10} - 6000 x^{9} + 108150 x^{8} - 144735 x^{7} + 809745 x^{6} - 1853951 x^{5} + 3501044 x^{4} - 6549159 x^{3} + 18283513 x^{2} - 17492952 x + 14031649 \)

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:  \(-9171686030885637573690636671=-\,19^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.38$
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:  $19, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(551=19\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{551}(1,·)$, $\chi_{551}(550,·)$, $\chi_{551}(265,·)$, $\chi_{551}(151,·)$, $\chi_{551}(303,·)$, $\chi_{551}(208,·)$, $\chi_{551}(210,·)$, $\chi_{551}(531,·)$, $\chi_{551}(20,·)$, $\chi_{551}(341,·)$, $\chi_{551}(343,·)$, $\chi_{551}(248,·)$, $\chi_{551}(400,·)$, $\chi_{551}(286,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{459414751598651181838292374837707824331569796599} a^{13} + \frac{166132028405411616153668177954680206299387028978}{459414751598651181838292374837707824331569796599} a^{12} + \frac{42601504408995429484411149570387020205501621835}{459414751598651181838292374837707824331569796599} a^{11} + \frac{159652733440659360007927377980143704948781636895}{459414751598651181838292374837707824331569796599} a^{10} - \frac{63177690426536109274033920052035758406582646120}{459414751598651181838292374837707824331569796599} a^{9} + \frac{208872437123861789064975122158494030447922884726}{459414751598651181838292374837707824331569796599} a^{8} - \frac{140991084702862478591239445023771477967282652777}{459414751598651181838292374837707824331569796599} a^{7} - \frac{61067516711108354831597264371228481774016542603}{459414751598651181838292374837707824331569796599} a^{6} + \frac{218164398864203476255286062560101729323741018993}{459414751598651181838292374837707824331569796599} a^{5} - \frac{97363055121809514257163886723435637408533842849}{459414751598651181838292374837707824331569796599} a^{4} - \frac{162074945260240209533496404650721732503657220674}{459414751598651181838292374837707824331569796599} a^{3} + \frac{219767788582630299352089080643326932022815053657}{459414751598651181838292374837707824331569796599} a^{2} - \frac{207113048828506073177839763659994517394203188851}{459414751598651181838292374837707824331569796599} a + \frac{224355527981534042153459141977568860604483015843}{459414751598651181838292374837707824331569796599}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{1846}$, which has order $14768$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-551}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$19$19.14.7.1$x^{14} - 109744 x^{8} + 3010936384 x^{2} - 14301947824$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$