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Properties

Label 14.0.13760207184...6523.2
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{13}$
Root discriminant $736.28$
Ramified primes $7, 29$
Class number $1248352$ (GRH)
Class group $[2, 2, 2, 156044]$ (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![97550940167, 136319294454, 64346564856, 11388585418, 959190225, 139144320, -13421345, -9384864, 5972260, -249690, -1218, -2233, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2233*x^11 - 1218*x^10 - 249690*x^9 + 5972260*x^8 - 9384864*x^7 - 13421345*x^6 + 139144320*x^5 + 959190225*x^4 + 11388585418*x^3 + 64346564856*x^2 + 136319294454*x + 97550940167)
 
gp: K = bnfinit(x^14 - 2233*x^11 - 1218*x^10 - 249690*x^9 + 5972260*x^8 - 9384864*x^7 - 13421345*x^6 + 139144320*x^5 + 959190225*x^4 + 11388585418*x^3 + 64346564856*x^2 + 136319294454*x + 97550940167, 1)
 

Normalized defining polynomial

\( x^{14} - 2233 x^{11} - 1218 x^{10} - 249690 x^{9} + 5972260 x^{8} - 9384864 x^{7} - 13421345 x^{6} + 139144320 x^{5} + 959190225 x^{4} + 11388585418 x^{3} + 64346564856 x^{2} + 136319294454 x + 97550940167 \)

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$$\chi_{1421}(1,·)$, $\chi_{1421}(1254,·)$, $\chi_{1421}(167,·)$, $\chi_{1421}(1420,·)$, $\chi_{1421}(846,·)$, $\chi_{1421}(1231,·)$, $\chi_{1421}(818,·)$, $\chi_{1421}(531,·)$, $\chi_{1421}(468,·)$, $\chi_{1421}(953,·)$, $\chi_{1421}(890,·)$, $\chi_{1421}(603,·)$, $\chi_{1421}(190,·)$, $\chi_{1421}(575,·)$$\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}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{13} - \frac{120923211754154264253435493049380589444578056164379571695215147793315759}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{12} - \frac{96561814084797200836787402532123010756080992040411308139077610370076864}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{11} - \frac{214043840412880394862491878508143496732827040112337657911298927095768660}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{10} - \frac{3164120954048646459104369033870184056378500034559928729416284948878879}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{9} + \frac{209731471825576027704612119525667888629952459590005149059307519586908891}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{8} - \frac{208339114876422830967601888412606541372562596819850044218299152295116369}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{7} + \frac{310166328066474174290382835218155107678440603112805916587766954403735084}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{6} - \frac{148618431216191690757343519238460969122223982705143697315574913420961455}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{5} - \frac{120181056907674871549083011578124474082355684973321076490854774746612138}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{4} + \frac{31607211957063992415363839845319732571953698283657807616255748218126}{79563169217148990426862329496658775395181883908608615004082563085741} a^{3} - \frac{94048892769351926723240248465790765448006117211962965599724450312490498}{638335306629186350194716469551693354995544254598766918177754403636900043} a^{2} + \frac{2238351734131629533961701651861874668595799365187499697580934038313763}{638335306629186350194716469551693354995544254598766918177754403636900043} a - \frac{226122964294278668833323434434815525719534096063221191695072918522278082}{638335306629186350194716469551693354995544254598766918177754403636900043}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{156044}$, which has order $1248352$ (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:  \( 93363968.92853843 \) (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

\(\Q(\sqrt{-203}) \), 7.7.8233120419813614521.5

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.25.95$x^{14} + 168 x^{13} + 140 x^{12} + 49 x^{11} + 49 x^{10} - 49 x^{9} - 168 x^{7} - 147 x^{6} + 147 x^{5} - 49 x^{4} - 98 x^{3} + 98 x^{2} - 98 x - 91$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.14.13.3$x^{14} - 464$$14$$1$$13$$C_{14}$$[\ ]_{14}$