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Properties

Label 14.0.74200338022...1232.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 29^{12}$
Root discriminant $50.70$
Ramified primes $2, 29$
Class number $344$ (GRH)
Class group $[2, 2, 86]$ (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![33251, -7326, 29677, -8292, 10130, -694, 2022, 152, 267, -104, 90, 14, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 9*x^12 + 14*x^11 + 90*x^10 - 104*x^9 + 267*x^8 + 152*x^7 + 2022*x^6 - 694*x^5 + 10130*x^4 - 8292*x^3 + 29677*x^2 - 7326*x + 33251)
 
gp: K = bnfinit(x^14 - 2*x^13 - 9*x^12 + 14*x^11 + 90*x^10 - 104*x^9 + 267*x^8 + 152*x^7 + 2022*x^6 - 694*x^5 + 10130*x^4 - 8292*x^3 + 29677*x^2 - 7326*x + 33251, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 9 x^{12} + 14 x^{11} + 90 x^{10} - 104 x^{9} + 267 x^{8} + 152 x^{7} + 2022 x^{6} - 694 x^{5} + 10130 x^{4} - 8292 x^{3} + 29677 x^{2} - 7326 x + 33251 \)

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:  \(-742003380228915810271232=-\,2^{21}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.70$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(232=2^{3}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{232}(1,·)$, $\chi_{232}(83,·)$, $\chi_{232}(227,·)$, $\chi_{232}(81,·)$, $\chi_{232}(65,·)$, $\chi_{232}(161,·)$, $\chi_{232}(139,·)$, $\chi_{232}(219,·)$, $\chi_{232}(49,·)$, $\chi_{232}(107,·)$, $\chi_{232}(59,·)$, $\chi_{232}(169,·)$, $\chi_{232}(25,·)$, $\chi_{232}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} - \frac{3}{17} a^{9} + \frac{1}{17} a^{8} + \frac{8}{17} a^{7} - \frac{2}{17} a^{6} - \frac{3}{17} a^{5} + \frac{1}{17} a^{4} - \frac{8}{17} a^{3} + \frac{1}{17} a^{2} + \frac{6}{17} a - \frac{2}{17}$, $\frac{1}{17} a^{11} - \frac{8}{17} a^{9} - \frac{6}{17} a^{8} + \frac{5}{17} a^{7} + \frac{8}{17} a^{6} - \frac{8}{17} a^{5} - \frac{5}{17} a^{4} - \frac{6}{17} a^{3} - \frac{8}{17} a^{2} - \frac{1}{17} a - \frac{6}{17}$, $\frac{1}{1003} a^{12} + \frac{24}{1003} a^{11} - \frac{409}{1003} a^{9} + \frac{430}{1003} a^{8} - \frac{352}{1003} a^{7} - \frac{342}{1003} a^{6} + \frac{7}{59} a^{5} + \frac{273}{1003} a^{4} + \frac{243}{1003} a^{3} + \frac{478}{1003} a^{2} - \frac{169}{1003} a - \frac{296}{1003}$, $\frac{1}{2229456199825211918387} a^{13} + \frac{863479743626220616}{2229456199825211918387} a^{12} + \frac{13896119066051190260}{2229456199825211918387} a^{11} + \frac{58321648288836955632}{2229456199825211918387} a^{10} + \frac{40635123044950587797}{131144482342659524611} a^{9} - \frac{648995865670570813567}{2229456199825211918387} a^{8} - \frac{935677769827102698}{3198645910796573771} a^{7} + \frac{714871024141475935259}{2229456199825211918387} a^{6} + \frac{461986596127080211028}{2229456199825211918387} a^{5} - \frac{818562890444413956960}{2229456199825211918387} a^{4} - \frac{137788194913693689037}{2229456199825211918387} a^{3} - \frac{491249711614941407496}{2229456199825211918387} a^{2} - \frac{387614244646881526282}{2229456199825211918387} a - \frac{20212568323509601324}{54376980483541754107}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{86}$, which has order $344$ (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:  \( 6020.98510015 \) (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

\(\Q(\sqrt{-2}) \), 7.7.594823321.1

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 R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$2$2.14.21.6$x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$