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Properties

Label 14.0.19995062629...7319.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,11^{7}\cdot 29^{13}$
Root discriminant $75.62$
Ramified primes $11, 29$
Class number $5120$ (GRH)
Class group $[2, 8, 8, 40]$ (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![741761, -1133182, 1166437, -306967, 252348, -189003, 68157, -25893, 17670, -1882, 1806, -75, 74, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 74*x^12 - 75*x^11 + 1806*x^10 - 1882*x^9 + 17670*x^8 - 25893*x^7 + 68157*x^6 - 189003*x^5 + 252348*x^4 - 306967*x^3 + 1166437*x^2 - 1133182*x + 741761)
 
gp: K = bnfinit(x^14 - x^13 + 74*x^12 - 75*x^11 + 1806*x^10 - 1882*x^9 + 17670*x^8 - 25893*x^7 + 68157*x^6 - 189003*x^5 + 252348*x^4 - 306967*x^3 + 1166437*x^2 - 1133182*x + 741761, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 74 x^{12} - 75 x^{11} + 1806 x^{10} - 1882 x^{9} + 17670 x^{8} - 25893 x^{7} + 68157 x^{6} - 189003 x^{5} + 252348 x^{4} - 306967 x^{3} + 1166437 x^{2} - 1133182 x + 741761 \)

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:  \(-199950626296934196778017319=-\,11^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.62$
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:  $11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(319=11\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{319}(1,·)$, $\chi_{319}(199,·)$, $\chi_{319}(296,·)$, $\chi_{319}(210,·)$, $\chi_{319}(45,·)$, $\chi_{319}(78,·)$, $\chi_{319}(109,·)$, $\chi_{319}(208,·)$, $\chi_{319}(241,·)$, $\chi_{319}(274,·)$, $\chi_{319}(23,·)$, $\chi_{319}(120,·)$, $\chi_{319}(111,·)$, $\chi_{319}(318,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} + \frac{6}{17} a^{9} + \frac{6}{17} a^{8} + \frac{8}{17} a^{7} + \frac{5}{17} a^{6} + \frac{1}{17} a^{5} + \frac{7}{17} a^{4} - \frac{2}{17} a^{3} + \frac{4}{17} a^{2} - \frac{7}{17} a$, $\frac{1}{17} a^{11} + \frac{4}{17} a^{9} + \frac{6}{17} a^{8} + \frac{8}{17} a^{7} + \frac{5}{17} a^{6} + \frac{1}{17} a^{5} + \frac{7}{17} a^{4} - \frac{1}{17} a^{3} + \frac{3}{17} a^{2} + \frac{8}{17} a$, $\frac{1}{41123} a^{12} - \frac{342}{41123} a^{11} + \frac{342}{41123} a^{10} - \frac{7324}{41123} a^{9} + \frac{2364}{41123} a^{8} - \frac{373}{1003} a^{7} + \frac{6339}{41123} a^{6} - \frac{17813}{41123} a^{5} + \frac{17090}{41123} a^{4} - \frac{552}{41123} a^{3} - \frac{15901}{41123} a^{2} + \frac{14142}{41123} a + \frac{910}{2419}$, $\frac{1}{453932205816323912771775575765689} a^{13} - \frac{3241545864512371210294240049}{453932205816323912771775575765689} a^{12} + \frac{939578144218307626034301932130}{453932205816323912771775575765689} a^{11} - \frac{5724678145183562042964308683134}{453932205816323912771775575765689} a^{10} + \frac{146721737369751038736647752150595}{453932205816323912771775575765689} a^{9} + \frac{4613864510021973356762872624451}{11071517215032290555409160384529} a^{8} + \frac{21016169604299479845683838842023}{453932205816323912771775575765689} a^{7} + \frac{18852173063594335176035586979646}{453932205816323912771775575765689} a^{6} - \frac{30523423398360247066816948670}{7693766200276676487657213148571} a^{5} + \frac{106444749402879399516379870881276}{453932205816323912771775575765689} a^{4} + \frac{146060283860839032197641567588214}{453932205816323912771775575765689} a^{3} + \frac{69631911095663556247735404260611}{453932205816323912771775575765689} a^{2} - \frac{107491127950799565584137694063604}{453932205816323912771775575765689} a - \frac{244296166150193131214696688668}{651265718531311209141715316737}$

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

Class group and class number

$C_{2}\times C_{8}\times C_{8}\times C_{40}$, which has order $5120$ (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

\(\Q(\sqrt{-319}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ R ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$11$11.14.7.1$x^{14} - 2662 x^{8} + 1771561 x^{2} - 311794736$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$