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Properties

Label 14.0.13760207184...6523.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{13}$
Root discriminant $736.28$
Ramified primes $7, 29$
Class number $307748$ (GRH)
Class group $[307748]$ (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![28996645309, 7015153824, 234668591136, 68261720093, 14862318170, 947498643, 342372898, 24872343, -429345, 30856, -18270, -609, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 609*x^11 - 18270*x^10 + 30856*x^9 - 429345*x^8 + 24872343*x^7 + 342372898*x^6 + 947498643*x^5 + 14862318170*x^4 + 68261720093*x^3 + 234668591136*x^2 + 7015153824*x + 28996645309)
 
gp: K = bnfinit(x^14 - 609*x^11 - 18270*x^10 + 30856*x^9 - 429345*x^8 + 24872343*x^7 + 342372898*x^6 + 947498643*x^5 + 14862318170*x^4 + 68261720093*x^3 + 234668591136*x^2 + 7015153824*x + 28996645309, 1)
 

Normalized defining polynomial

\( x^{14} - 609 x^{11} - 18270 x^{10} + 30856 x^{9} - 429345 x^{8} + 24872343 x^{7} + 342372898 x^{6} + 947498643 x^{5} + 14862318170 x^{4} + 68261720093 x^{3} + 234668591136 x^{2} + 7015153824 x + 28996645309 \)

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:  \(-13760207184971837367389621368916758066523=-\,7^{25}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $736.28$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1,·)$, $\chi_{1421}(34,·)$, $\chi_{1421}(1156,·)$, $\chi_{1421}(265,·)$, $\chi_{1421}(1387,·)$, $\chi_{1421}(1420,·)$, $\chi_{1421}(825,·)$, $\chi_{1421}(209,·)$, $\chi_{1421}(370,·)$, $\chi_{1421}(596,·)$, $\chi_{1421}(937,·)$, $\chi_{1421}(484,·)$, $\chi_{1421}(1051,·)$, $\chi_{1421}(1212,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{9} - \frac{1}{5} a^{3} + \frac{1}{5}$, $\frac{1}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{13} + \frac{377364444386716025577117209163027325076055488524959287187433606647096523}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{12} - \frac{39792440114637323548408743217534305025356579036155810705754790822820116}{793194673815321916168802209881571514094040309271482588561688082412573727} a^{11} + \frac{346695664761126075788516611846684944123722786755023770224700845522625837}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{10} + \frac{704127674303767302282835592948325161210758888668988445341205135508425813}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{9} + \frac{1148521732371299334230348617460090556699852473445895766225252870826384152}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{8} + \frac{1094494364397275407603264928544069184445219121859449083422544909355331197}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{7} - \frac{924574749805500006964403101540695976974117376765000059930459840984187559}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{6} - \frac{146562404710353505380879940405758378826471702871173254424310220137416495}{793194673815321916168802209881571514094040309271482588561688082412573727} a^{5} + \frac{1466450487987089331120703670533187694527217172164977560014318975069495501}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{4} + \frac{1458416156533339909376468497417755200967813741021140513179704837486133779}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{3} + \frac{1194611516908542848934919402824324210006529318524635243266651060234385986}{3965973369076609580844011049407857570470201546357412942808440412062868635} a^{2} - \frac{1027199290778459089158173143072710234904337664285183229246173559949358003}{3965973369076609580844011049407857570470201546357412942808440412062868635} a + \frac{553879874654705716933925165906088282821374653099325721101056005598836656}{3965973369076609580844011049407857570470201546357412942808440412062868635}$

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

Class group and class number

$C_{307748}$, which has order $307748$ (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:  \( 368248548.12946504 \) (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

\(\Q(\sqrt{-203}) \), 7.7.8233120419813614521.2

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.72$x^{14} + 63 x^{13} + 35 x^{12} + 98 x^{11} + 98 x^{10} - 98 x^{9} + 147 x^{8} - 112 x^{7} - 49 x^{6} + 147 x^{5} + 98 x^{4} - 49 x^{3} + 147 x^{2} + 49 x + 63$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.14.13.6$x^{14} - 29696$$14$$1$$13$$C_{14}$$[\ ]_{14}$