↑

Properties

Label 14.0.32918715676...4443.3
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 43^{13}$
Root discriminant $923.70$
Ramified primes $7, 43$
Class number $3112193$ (GRH)
Class group $[3112193]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11968080874901, 406889558152, 5366060727678, 1283975888426, 75633303333, -2202387502, -85626373, 4397266, 6674976, 131838, 39130, -1505, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 1505*x^11 + 39130*x^10 + 131838*x^9 + 6674976*x^8 + 4397266*x^7 - 85626373*x^6 - 2202387502*x^5 + 75633303333*x^4 + 1283975888426*x^3 + 5366060727678*x^2 + 406889558152*x + 11968080874901)
 
gp: K = bnfinit(x^14 - 1505*x^11 + 39130*x^10 + 131838*x^9 + 6674976*x^8 + 4397266*x^7 - 85626373*x^6 - 2202387502*x^5 + 75633303333*x^4 + 1283975888426*x^3 + 5366060727678*x^2 + 406889558152*x + 11968080874901, 1)
 

Normalized defining polynomial

\( x^{14} - 1505 x^{11} + 39130 x^{10} + 131838 x^{9} + 6674976 x^{8} + 4397266 x^{7} - 85626373 x^{6} - 2202387502 x^{5} + 75633303333 x^{4} + 1283975888426 x^{3} + 5366060727678 x^{2} + 406889558152 x + 11968080874901 \)

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:  \(-329187156767051876908796215815371233484443=-\,7^{24}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $923.70$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(1,·)$, $\chi_{2107}(323,·)$, $\chi_{2107}(1415,·)$, $\chi_{2107}(778,·)$, $\chi_{2107}(204,·)$, $\chi_{2107}(1933,·)$, $\chi_{2107}(1583,·)$, $\chi_{2107}(561,·)$, $\chi_{2107}(309,·)$, $\chi_{2107}(1016,·)$, $\chi_{2107}(666,·)$, $\chi_{2107}(687,·)$, $\chi_{2107}(1086,·)$, $\chi_{2107}(575,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23} a^{10} + \frac{1}{23} a^{9} - \frac{4}{23} a^{8} + \frac{9}{23} a^{7} - \frac{8}{23} a^{6} - \frac{5}{23} a^{5} + \frac{8}{23} a^{4} + \frac{10}{23} a^{2} + \frac{11}{23} a$, $\frac{1}{23} a^{11} - \frac{5}{23} a^{9} - \frac{10}{23} a^{8} + \frac{6}{23} a^{7} + \frac{3}{23} a^{6} - \frac{10}{23} a^{5} - \frac{8}{23} a^{4} + \frac{10}{23} a^{3} + \frac{1}{23} a^{2} - \frac{11}{23} a$, $\frac{1}{41791} a^{12} - \frac{463}{41791} a^{11} - \frac{459}{41791} a^{10} - \frac{610}{41791} a^{9} + \frac{15583}{41791} a^{8} + \frac{2615}{41791} a^{7} + \frac{18379}{41791} a^{6} - \frac{4585}{41791} a^{5} + \frac{10271}{41791} a^{4} - \frac{2421}{41791} a^{3} - \frac{570}{1817} a^{2} - \frac{6387}{41791} a - \frac{6}{23}$, $\frac{1}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{13} - \frac{1309940810885527685973486636478240466328349987282002447559404443016285}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{12} - \frac{21197698679078907105797169735386503652857533689897747462264971482970774145}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{11} + \frac{42993608121049789064701721682959217228976232006175269960959897111722214873}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{10} + \frac{635480695856503839208173027410898160894114747584918468946514228423816195896}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{9} + \frac{186892594141512313093218114674206465462942834303270946927919573302933790539}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{8} - \frac{955534485674261251976513731125361983614861525581887003391012090017559878610}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{7} + \frac{397411910207004640204510953992508216997525083181994410274395672048045311181}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{6} - \frac{837493027017909247513398457620755531170775372997950128859535556512695966278}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{5} + \frac{508641279640415779109629723787524137779040065335907286935795310431524309580}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{4} - \frac{1204189657223799163663934070802823952621757548937106211796857742325687835461}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{3} - \frac{585371401684783063661395065507919765621367817181445898545972702337908677410}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a^{2} - \frac{985208508993073966733621785237328432441925663348664645264605799173308136075}{2542095299757409057453888961018854217354108514506032560150907721624448579979} a + \frac{36819265452047803433389699896586473656795351184469077386208785670103959}{1399061805039850884674677468915164676584539633740249069978485262313950787}$

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

Class group and class number

$C_{3112193}$, which has order $3112193$ (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:  \( 301031344.50909585 \) (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

\(\Q(\sqrt{-43}) \), 7.7.87495801462998035849.5

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$43$43.14.13.10$x^{14} + 2539107$$14$$1$$13$$C_{14}$$[\ ]_{14}$