↑

Properties

Label 14.0.31388668949...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 5^{7}\cdot 7^{24}$
Root discriminant $177.73$
Ramified primes $2, 5, 7$
Class number $283234$ (GRH)
Class group $[283234]$ (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![178610849, 47642756, 48628468, 8567650, 6352920, 804706, 508025, 33840, 27041, 686, 1043, -42, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 28*x^12 - 42*x^11 + 1043*x^10 + 686*x^9 + 27041*x^8 + 33840*x^7 + 508025*x^6 + 804706*x^5 + 6352920*x^4 + 8567650*x^3 + 48628468*x^2 + 47642756*x + 178610849)
 
gp: K = bnfinit(x^14 + 28*x^12 - 42*x^11 + 1043*x^10 + 686*x^9 + 27041*x^8 + 33840*x^7 + 508025*x^6 + 804706*x^5 + 6352920*x^4 + 8567650*x^3 + 48628468*x^2 + 47642756*x + 178610849, 1)
 

Normalized defining polynomial

\( x^{14} + 28 x^{12} - 42 x^{11} + 1043 x^{10} + 686 x^{9} + 27041 x^{8} + 33840 x^{7} + 508025 x^{6} + 804706 x^{5} + 6352920 x^{4} + 8567650 x^{3} + 48628468 x^{2} + 47642756 x + 178610849 \)

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:  \(-31388668949392001335459840000000=-\,2^{21}\cdot 5^{7}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.73$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1960=2^{3}\cdot 5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1960}(1,·)$, $\chi_{1960}(1219,·)$, $\chi_{1960}(1121,·)$, $\chi_{1960}(841,·)$, $\chi_{1960}(939,·)$, $\chi_{1960}(1681,·)$, $\chi_{1960}(99,·)$, $\chi_{1960}(659,·)$, $\chi_{1960}(561,·)$, $\chi_{1960}(1779,·)$, $\chi_{1960}(281,·)$, $\chi_{1960}(379,·)$, $\chi_{1960}(1499,·)$, $\chi_{1960}(1401,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9} - \frac{1}{19} a^{8} - \frac{8}{19} a^{7} - \frac{6}{19} a^{6} + \frac{9}{19} a^{5} - \frac{4}{19} a^{4} + \frac{3}{19} a^{3} - \frac{5}{19} a$, $\frac{1}{19} a^{10} - \frac{9}{19} a^{8} + \frac{5}{19} a^{7} + \frac{3}{19} a^{6} + \frac{5}{19} a^{5} - \frac{1}{19} a^{4} + \frac{3}{19} a^{3} - \frac{5}{19} a^{2} - \frac{5}{19} a$, $\frac{1}{19} a^{11} - \frac{4}{19} a^{8} + \frac{7}{19} a^{7} + \frac{8}{19} a^{6} + \frac{4}{19} a^{5} + \frac{5}{19} a^{4} + \frac{3}{19} a^{3} - \frac{5}{19} a^{2} - \frac{7}{19} a$, $\frac{1}{589} a^{12} + \frac{13}{589} a^{11} + \frac{13}{589} a^{10} - \frac{2}{589} a^{9} - \frac{259}{589} a^{8} + \frac{167}{589} a^{7} - \frac{74}{589} a^{6} + \frac{197}{589} a^{5} + \frac{85}{589} a^{4} + \frac{174}{589} a^{3} - \frac{175}{589} a^{2} - \frac{166}{589} a + \frac{12}{31}$, $\frac{1}{101217628958112688241068370459905031} a^{13} - \frac{47828087202342698436619966261840}{101217628958112688241068370459905031} a^{12} - \frac{1726974257978612971731698532375234}{101217628958112688241068370459905031} a^{11} - \frac{1188971992297651048742931558622991}{101217628958112688241068370459905031} a^{10} + \frac{543817830569803930077085669477506}{101217628958112688241068370459905031} a^{9} + \frac{13244358111356022711527424849505904}{101217628958112688241068370459905031} a^{8} - \frac{5427564216369252609616730520022158}{101217628958112688241068370459905031} a^{7} + \frac{13569097779823773904585787165897326}{101217628958112688241068370459905031} a^{6} - \frac{28561406466862370282815181312251321}{101217628958112688241068370459905031} a^{5} + \frac{46188716065018899065274876320402608}{101217628958112688241068370459905031} a^{4} - \frac{20177624259542231227335648286389346}{101217628958112688241068370459905031} a^{3} - \frac{27473418593305594232597018957980828}{101217628958112688241068370459905031} a^{2} + \frac{27377325100066835634857711562605265}{101217628958112688241068370459905031} a - \frac{9219380381107844996195282893176}{171846568689495226215735773276579}$

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

Class group and class number

$C_{283234}$, which has order $283234$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-10}) \), 7.7.13841287201.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$2.14.21.40$x^{14} + 8 x^{13} - 4 x^{12} + 4 x^{11} + 5 x^{10} - 4 x^{9} - 4 x^{8} + 2 x^{7} - x^{6} + 6 x^{5} - 4 x^{4} + 6 x^{3} + 3 x^{2} + 6 x + 3$$2$$7$$21$$C_{14}$$[3]^{7}$
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$