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Properties

Label 14.0.75966758316...2827.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 239^{12}$
Root discriminant $189.32$
Ramified primes $3, 239$
Class number $7624$ (GRH)
Class group $[2, 2, 1906]$ (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![26863489, 46299739, 74729515, 27913574, 18493219, 2203236, 2712410, 164988, 207881, -17168, 8749, -288, 103, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 103*x^12 - 288*x^11 + 8749*x^10 - 17168*x^9 + 207881*x^8 + 164988*x^7 + 2712410*x^6 + 2203236*x^5 + 18493219*x^4 + 27913574*x^3 + 74729515*x^2 + 46299739*x + 26863489)
 
gp: K = bnfinit(x^14 - x^13 + 103*x^12 - 288*x^11 + 8749*x^10 - 17168*x^9 + 207881*x^8 + 164988*x^7 + 2712410*x^6 + 2203236*x^5 + 18493219*x^4 + 27913574*x^3 + 74729515*x^2 + 46299739*x + 26863489, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 103 x^{12} - 288 x^{11} + 8749 x^{10} - 17168 x^{9} + 207881 x^{8} + 164988 x^{7} + 2712410 x^{6} + 2203236 x^{5} + 18493219 x^{4} + 27913574 x^{3} + 74729515 x^{2} + 46299739 x + 26863489 \)

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:  \(-75966758316706681853567921242827=-\,3^{7}\cdot 239^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $189.32$
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:  $3, 239$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(717=3\cdot 239\)
Dirichlet character group:    $\lbrace$$\chi_{717}(1,·)$, $\chi_{717}(98,·)$, $\chi_{717}(100,·)$, $\chi_{717}(263,·)$, $\chi_{717}(488,·)$, $\chi_{717}(10,·)$, $\chi_{717}(679,·)$, $\chi_{717}(44,·)$, $\chi_{717}(578,·)$, $\chi_{717}(337,·)$, $\chi_{717}(502,·)$, $\chi_{717}(440,·)$, $\chi_{717}(283,·)$, $\chi_{717}(479,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23} a^{10} + \frac{10}{23} a^{9} + \frac{4}{23} a^{8} + \frac{10}{23} a^{7} + \frac{6}{23} a^{6} + \frac{6}{23} a^{5} + \frac{9}{23} a^{3} + \frac{11}{23} a^{2} + \frac{4}{23} a + \frac{4}{23}$, $\frac{1}{23} a^{11} - \frac{4}{23} a^{9} - \frac{7}{23} a^{8} - \frac{2}{23} a^{7} - \frac{8}{23} a^{6} + \frac{9}{23} a^{5} + \frac{9}{23} a^{4} - \frac{10}{23} a^{3} + \frac{9}{23} a^{2} + \frac{10}{23} a + \frac{6}{23}$, $\frac{1}{10323343} a^{12} - \frac{110647}{10323343} a^{11} + \frac{110749}{10323343} a^{10} + \frac{4562766}{10323343} a^{9} - \frac{4924956}{10323343} a^{8} + \frac{4955134}{10323343} a^{7} + \frac{88467}{10323343} a^{6} - \frac{4445720}{10323343} a^{5} - \frac{4061917}{10323343} a^{4} - \frac{3777775}{10323343} a^{3} + \frac{4718280}{10323343} a^{2} + \frac{2463237}{10323343} a - \frac{2030878}{10323343}$, $\frac{1}{24722886956523247620924040592654236767252053} a^{13} + \frac{1062919841657605250515455631474939377}{24722886956523247620924040592654236767252053} a^{12} + \frac{393285564892727247221822975595656650536941}{24722886956523247620924040592654236767252053} a^{11} + \frac{135000003743933749535357074251385887481902}{24722886956523247620924040592654236767252053} a^{10} - \frac{3441112035619765007182222709814480754572809}{24722886956523247620924040592654236767252053} a^{9} + \frac{2807374664134980397008599956883230887751360}{24722886956523247620924040592654236767252053} a^{8} + \frac{328225971558212577690974566898844573749899}{24722886956523247620924040592654236767252053} a^{7} + \frac{518433011910074828765025186469603207441565}{24722886956523247620924040592654236767252053} a^{6} - \frac{8511996367323304318745948342777053243524611}{24722886956523247620924040592654236767252053} a^{5} + \frac{429646499630514664730295547757087647250597}{1074908128544489026996697417071923337706611} a^{4} - \frac{8154105372153990323863100853656512436994197}{24722886956523247620924040592654236767252053} a^{3} + \frac{2515972353176636396721633732715706169209781}{24722886956523247620924040592654236767252053} a^{2} + \frac{8249743103808191552684529420703729040381055}{24722886956523247620924040592654236767252053} a - \frac{436655642981946701268355560477836734316}{4769995554027252097419263089456730998891}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1906}$, which has order $7624$ (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:  \( -\frac{51681703564915017913449529854}{2394852806549510911428985803596202971} a^{13} + \frac{46843315289024321442656022402}{2394852806549510911428985803596202971} a^{12} - \frac{5358650797744434753172185991476}{2394852806549510911428985803596202971} a^{11} + \frac{14573682317120861823764732798125}{2394852806549510911428985803596202971} a^{10} - \frac{454792850258874498675525989963197}{2394852806549510911428985803596202971} a^{9} + \frac{871051472790390299538871409824021}{2394852806549510911428985803596202971} a^{8} - \frac{11030316836184649876375740931857632}{2394852806549510911428985803596202971} a^{7} - \frac{7656329132663771064524569747070397}{2394852806549510911428985803596202971} a^{6} - \frac{149692132002401055709378168984894288}{2394852806549510911428985803596202971} a^{5} - \frac{109313984993274354294755745336581708}{2394852806549510911428985803596202971} a^{4} - \frac{1016230578137625732262077417341134849}{2394852806549510911428985803596202971} a^{3} - \frac{1112072949085062945830542153982085523}{2394852806549510911428985803596202971} a^{2} - \frac{4211182356954355500384202450813021931}{2394852806549510911428985803596202971} a - \frac{42375743076623400257224069730742}{462059194780920492268760525486437} \) (order $6$)
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:  \( 3022802.0673343516 \) (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

\(\Q(\sqrt{-3}) \), 7.7.186374892382561.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
239Data not computed