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Properties

Label 14.0.90901537886...2912.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 113^{12}$
Root discriminant $162.68$
Ramified primes $2, 113$
Class number $11368$ (GRH)
Class group $[2, 14, 406]$ (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![1550827, -76170, 1757945, -64960, 660458, -12686, 53766, 9188, -10545, 3644, 2330, -2, -81, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 81*x^12 - 2*x^11 + 2330*x^10 + 3644*x^9 - 10545*x^8 + 9188*x^7 + 53766*x^6 - 12686*x^5 + 660458*x^4 - 64960*x^3 + 1757945*x^2 - 76170*x + 1550827)
 
gp: K = bnfinit(x^14 - 2*x^13 - 81*x^12 - 2*x^11 + 2330*x^10 + 3644*x^9 - 10545*x^8 + 9188*x^7 + 53766*x^6 - 12686*x^5 + 660458*x^4 - 64960*x^3 + 1757945*x^2 - 76170*x + 1550827, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 81 x^{12} - 2 x^{11} + 2330 x^{10} + 3644 x^{9} - 10545 x^{8} + 9188 x^{7} + 53766 x^{6} - 12686 x^{5} + 660458 x^{4} - 64960 x^{3} + 1757945 x^{2} - 76170 x + 1550827 \)

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:  \(-9090153788613196226613752102912=-\,2^{21}\cdot 113^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $162.68$
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:  $2, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(904=2^{3}\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{904}(897,·)$, $\chi_{904}(227,·)$, $\chi_{904}(1,·)$, $\chi_{904}(129,·)$, $\chi_{904}(355,·)$, $\chi_{904}(369,·)$, $\chi_{904}(49,·)$, $\chi_{904}(595,·)$, $\chi_{904}(275,·)$, $\chi_{904}(561,·)$, $\chi_{904}(593,·)$, $\chi_{904}(787,·)$, $\chi_{904}(819,·)$, $\chi_{904}(219,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{73} a^{11} - \frac{5}{73} a^{10} - \frac{23}{73} a^{9} - \frac{32}{73} a^{8} - \frac{19}{73} a^{7} + \frac{22}{73} a^{6} - \frac{29}{73} a^{5} - \frac{13}{73} a^{4} - \frac{5}{73} a^{3} - \frac{23}{73} a^{2} - \frac{21}{73} a - \frac{36}{73}$, $\frac{1}{1501391} a^{12} - \frac{6487}{1501391} a^{11} - \frac{638191}{1501391} a^{10} - \frac{11181}{1501391} a^{9} + \frac{486338}{1501391} a^{8} - \frac{143489}{1501391} a^{7} - \frac{375576}{1501391} a^{6} - \frac{356834}{1501391} a^{5} + \frac{488900}{1501391} a^{4} - \frac{146609}{1501391} a^{3} + \frac{215641}{1501391} a^{2} + \frac{91848}{1501391} a + \frac{499364}{1501391}$, $\frac{1}{1907849175547288212935283693167} a^{13} + \frac{270489052969339792914638}{1907849175547288212935283693167} a^{12} - \frac{6843344459968197543049410111}{1907849175547288212935283693167} a^{11} + \frac{845680876416276765154449377287}{1907849175547288212935283693167} a^{10} - \frac{446198689760645482880869131270}{1907849175547288212935283693167} a^{9} - \frac{949605147667366531117747281161}{1907849175547288212935283693167} a^{8} - \frac{176038965834461499472789783854}{1907849175547288212935283693167} a^{7} - \frac{406463711940527321946759711784}{1907849175547288212935283693167} a^{6} + \frac{351307486505224339040902605978}{1907849175547288212935283693167} a^{5} + \frac{684402737259142813890140771843}{1907849175547288212935283693167} a^{4} + \frac{243742213206437668177337175464}{1907849175547288212935283693167} a^{3} - \frac{777886619558570064916178271470}{1907849175547288212935283693167} a^{2} + \frac{850556261298288848407836606094}{1907849175547288212935283693167} a + \frac{112788482963400129695525682454}{1907849175547288212935283693167}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{406}$, which has order $11368$ (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:  \( 222748.97284811488 \) (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

\(\Q(\sqrt{-2}) \), 7.7.2081951752609.1

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 R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$2$2.14.21.6$x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$113$113.7.6.1$x^{7} - 113$$7$$1$$6$$C_7$$[\ ]_{7}$
113.7.6.1$x^{7} - 113$$7$$1$$6$$C_7$$[\ ]_{7}$