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Properties

Label 14.0.53005949918...3307.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 281^{12}$
Root discriminant $217.50$
Ramified primes $3, 281$
Class number $50723$ (GRH)
Class group $[50723]$ (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![117649, 982009, 8867677, -5062204, 5826655, -2578842, 2348246, -1029384, 415211, -88844, 15895, -1302, 121, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 121*x^12 - 1302*x^11 + 15895*x^10 - 88844*x^9 + 415211*x^8 - 1029384*x^7 + 2348246*x^6 - 2578842*x^5 + 5826655*x^4 - 5062204*x^3 + 8867677*x^2 + 982009*x + 117649)
 
gp: K = bnfinit(x^14 - x^13 + 121*x^12 - 1302*x^11 + 15895*x^10 - 88844*x^9 + 415211*x^8 - 1029384*x^7 + 2348246*x^6 - 2578842*x^5 + 5826655*x^4 - 5062204*x^3 + 8867677*x^2 + 982009*x + 117649, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 121 x^{12} - 1302 x^{11} + 15895 x^{10} - 88844 x^{9} + 415211 x^{8} - 1029384 x^{7} + 2348246 x^{6} - 2578842 x^{5} + 5826655 x^{4} - 5062204 x^{3} + 8867677 x^{2} + 982009 x + 117649 \)

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:  \(-530059499186032039461665494643307=-\,3^{7}\cdot 281^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $217.50$
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:  $3, 281$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(843=3\cdot 281\)
Dirichlet character group:    $\lbrace$$\chi_{843}(1,·)$, $\chi_{843}(743,·)$, $\chi_{843}(811,·)$, $\chi_{843}(109,·)$, $\chi_{843}(79,·)$, $\chi_{843}(530,·)$, $\chi_{843}(563,·)$, $\chi_{843}(340,·)$, $\chi_{843}(181,·)$, $\chi_{843}(727,·)$, $\chi_{843}(641,·)$, $\chi_{843}(59,·)$, $\chi_{843}(446,·)$, $\chi_{843}(671,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{49} a^{11} + \frac{1}{49} a^{10} - \frac{3}{49} a^{9} + \frac{1}{49} a^{8} + \frac{3}{7} a^{6} - \frac{22}{49} a^{5} + \frac{13}{49} a^{4} + \frac{24}{49} a^{3} - \frac{8}{49} a^{2} + \frac{3}{7} a$, $\frac{1}{1692335981} a^{12} - \frac{6250557}{1692335981} a^{11} + \frac{107378374}{1692335981} a^{10} - \frac{66958369}{1692335981} a^{9} + \frac{81175837}{1692335981} a^{8} - \frac{7225361}{241762283} a^{7} - \frac{105683789}{1692335981} a^{6} - \frac{757679736}{1692335981} a^{5} + \frac{233453999}{1692335981} a^{4} + \frac{661028552}{1692335981} a^{3} + \frac{123869234}{1692335981} a^{2} + \frac{40332231}{241762283} a - \frac{3661122}{34537469}$, $\frac{1}{10696373907890656960189388119619393} a^{13} - \frac{1122689301785156274398972}{10696373907890656960189388119619393} a^{12} - \frac{88882195761083722971882880155105}{10696373907890656960189388119619393} a^{11} - \frac{80427650743325635197390001969753}{1528053415412950994312769731374199} a^{10} - \frac{289585905880589755845682400487909}{10696373907890656960189388119619393} a^{9} - \frac{75626237300524363677132025337690}{1528053415412950994312769731374199} a^{8} - \frac{224270952771949713814109475082079}{10696373907890656960189388119619393} a^{7} - \frac{2297584783097220658699639689163063}{10696373907890656960189388119619393} a^{6} + \frac{830769431512096291648946679349643}{10696373907890656960189388119619393} a^{5} - \frac{254584793956414862668856819019129}{1528053415412950994312769731374199} a^{4} - \frac{947901203666460279041275310830484}{10696373907890656960189388119619393} a^{3} - \frac{316230364385492218243023355429959}{1528053415412950994312769731374199} a^{2} - \frac{107281682745690520232846167190590}{218293345058992999187538533053457} a - \frac{27823376680720127383365248507}{4454966225693734677296704756193}$

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

Class group and class number

$C_{50723}$, which has order $50723$ (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:  \( \frac{41047875211280137134414}{43048781982165552358985105383} a^{13} - \frac{42800265099002327660706}{43048781982165552358985105383} a^{12} + \frac{4962264652431790677020538}{43048781982165552358985105383} a^{11} - \frac{7665057289429301563615613}{6149825997452221765569300769} a^{10} + \frac{653989859515508178119719121}{43048781982165552358985105383} a^{9} - \frac{523886131819688898997162577}{6149825997452221765569300769} a^{8} + \frac{17107110675144800963289790198}{43048781982165552358985105383} a^{7} - \frac{42519357059736019353796126791}{43048781982165552358985105383} a^{6} + \frac{96187269316525565567228976758}{43048781982165552358985105383} a^{5} - \frac{15225125216741658898937526134}{6149825997452221765569300769} a^{4} + \frac{238224539504723863706578460813}{43048781982165552358985105383} a^{3} - \frac{32202313339926738877606822393}{6149825997452221765569300769} a^{2} + \frac{7354632194762661199377762007}{878546571064603109367042967} a + \frac{16620564088037966768368123}{17929521858461287946266183} \) (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:  \( 12176100.831221418 \) (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

\(\Q(\sqrt{-3}) \), 7.7.492309163417681.1

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 ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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}$
281Data not computed