↑

Properties

Label 14.0.83801419645...9952.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 43^{12}$
Root discriminant $71.07$
Ramified primes $2, 43$
Class number $203$ (GRH)
Class group $[203]$ (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![208427, -210818, 124449, -70268, 57742, -26778, 4950, -824, 2063, -1164, 138, 82, -21, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 21*x^12 + 82*x^11 + 138*x^10 - 1164*x^9 + 2063*x^8 - 824*x^7 + 4950*x^6 - 26778*x^5 + 57742*x^4 - 70268*x^3 + 124449*x^2 - 210818*x + 208427)
 
gp: K = bnfinit(x^14 - 2*x^13 - 21*x^12 + 82*x^11 + 138*x^10 - 1164*x^9 + 2063*x^8 - 824*x^7 + 4950*x^6 - 26778*x^5 + 57742*x^4 - 70268*x^3 + 124449*x^2 - 210818*x + 208427, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 21 x^{12} + 82 x^{11} + 138 x^{10} - 1164 x^{9} + 2063 x^{8} - 824 x^{7} + 4950 x^{6} - 26778 x^{5} + 57742 x^{4} - 70268 x^{3} + 124449 x^{2} - 210818 x + 208427 \)

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:  \(-83801419645740806624509952=-\,2^{21}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.07$
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:  $2, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(344=2^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{344}(107,·)$, $\chi_{344}(1,·)$, $\chi_{344}(259,·)$, $\chi_{344}(97,·)$, $\chi_{344}(145,·)$, $\chi_{344}(41,·)$, $\chi_{344}(193,·)$, $\chi_{344}(11,·)$, $\chi_{344}(305,·)$, $\chi_{344}(35,·)$, $\chi_{344}(219,·)$, $\chi_{344}(121,·)$, $\chi_{344}(59,·)$, $\chi_{344}(299,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{302992315809994592990865223} a^{13} + \frac{3171984744725515900345099}{302992315809994592990865223} a^{12} + \frac{16877315909169163584113163}{302992315809994592990865223} a^{11} + \frac{679515527388687626922924}{43284616544284941855837889} a^{10} + \frac{64665629537667958930291338}{302992315809994592990865223} a^{9} - \frac{140186941182308219127992963}{302992315809994592990865223} a^{8} + \frac{139638096696426010055194868}{302992315809994592990865223} a^{7} + \frac{18505913968174287132433581}{302992315809994592990865223} a^{6} + \frac{22670395739979017831894987}{302992315809994592990865223} a^{5} + \frac{81357402527848260757631826}{302992315809994592990865223} a^{4} - \frac{66677428295857930395994426}{302992315809994592990865223} a^{3} - \frac{51594119512712243254222730}{302992315809994592990865223} a^{2} - \frac{67319788757898807659077814}{302992315809994592990865223} a - \frac{271715657191780180001930}{1178958427276243552493639}$

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

Class group and class number

$C_{203}$, which has order $203$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-2}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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}$$[3]^{7}$
$43$43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$