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Properties

Label 14.0.23133267811...1363.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{7}$
Root discriminant $173.90$
Ramified primes $7, 29$
Class number $405332$ (GRH)
Class group $[405332]$ (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![196014583, -188014449, 189410389, -162190658, 117948719, -55907334, 16797641, -4272939, 941430, -107492, 26467, -665, 294, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 294*x^12 - 665*x^11 + 26467*x^10 - 107492*x^9 + 941430*x^8 - 4272939*x^7 + 16797641*x^6 - 55907334*x^5 + 117948719*x^4 - 162190658*x^3 + 189410389*x^2 - 188014449*x + 196014583)
 
gp: K = bnfinit(x^14 + 294*x^12 - 665*x^11 + 26467*x^10 - 107492*x^9 + 941430*x^8 - 4272939*x^7 + 16797641*x^6 - 55907334*x^5 + 117948719*x^4 - 162190658*x^3 + 189410389*x^2 - 188014449*x + 196014583, 1)
 

Normalized defining polynomial

\( x^{14} + 294 x^{12} - 665 x^{11} + 26467 x^{10} - 107492 x^{9} + 941430 x^{8} - 4272939 x^{7} + 16797641 x^{6} - 55907334 x^{5} + 117948719 x^{4} - 162190658 x^{3} + 189410389 x^{2} - 188014449 x + 196014583 \)

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:  \(-23133267811084759683438204281363=-\,7^{25}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $173.90$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(608,·)$, $\chi_{1421}(1,·)$, $\chi_{1421}(610,·)$, $\chi_{1421}(1219,·)$, $\chi_{1421}(1217,·)$, $\chi_{1421}(204,·)$, $\chi_{1421}(202,·)$, $\chi_{1421}(811,·)$, $\chi_{1421}(1420,·)$, $\chi_{1421}(813,·)$, $\chi_{1421}(405,·)$, $\chi_{1421}(1014,·)$, $\chi_{1421}(407,·)$, $\chi_{1421}(1016,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9} - \frac{2}{19} a^{8} + \frac{7}{19} a^{7} - \frac{7}{19} a^{6} + \frac{2}{19} a^{5} - \frac{7}{19} a^{4} + \frac{3}{19} a^{3} - \frac{3}{19} a^{2} + \frac{6}{19} a$, $\frac{1}{589} a^{10} + \frac{10}{589} a^{9} + \frac{211}{589} a^{8} + \frac{96}{589} a^{7} + \frac{89}{589} a^{6} + \frac{245}{589} a^{5} - \frac{43}{589} a^{4} + \frac{52}{589} a^{3} + \frac{8}{589} a^{2} + \frac{167}{589} a + \frac{7}{31}$, $\frac{1}{589} a^{11} - \frac{13}{589} a^{9} + \frac{1}{589} a^{8} + \frac{28}{589} a^{7} + \frac{223}{589} a^{6} + \frac{204}{589} a^{5} + \frac{172}{589} a^{4} + \frac{294}{589} a^{3} - \frac{130}{589} a^{2} + \frac{75}{589} a - \frac{8}{31}$, $\frac{1}{589} a^{12} + \frac{7}{589} a^{9} + \frac{74}{589} a^{8} + \frac{14}{589} a^{7} - \frac{127}{589} a^{6} + \frac{164}{589} a^{5} + \frac{14}{589} a^{4} + \frac{174}{589} a^{3} - \frac{2}{31} a^{2} + \frac{97}{589} a - \frac{2}{31}$, $\frac{1}{3168500939917812167253300837130397521551187129643} a^{13} + \frac{76488610028974934957930879241584287900212538}{3168500939917812167253300837130397521551187129643} a^{12} + \frac{1560084491484226669189228867990350630978808393}{3168500939917812167253300837130397521551187129643} a^{11} - \frac{1782356513088274247946735212333248505411424071}{3168500939917812167253300837130397521551187129643} a^{10} - \frac{6513798104157158893715163685488658293766598422}{3168500939917812167253300837130397521551187129643} a^{9} + \frac{96120657740534483971209054867651875835503258357}{3168500939917812167253300837130397521551187129643} a^{8} - \frac{48784849597666630872438577728591079080467262567}{166763207364095377223857938796336711660588796297} a^{7} - \frac{115405424900852548878935716911764936136029619588}{3168500939917812167253300837130397521551187129643} a^{6} + \frac{191322408886293555100732812120667644713535654649}{3168500939917812167253300837130397521551187129643} a^{5} - \frac{763624456167592066659471741026180783423296653379}{3168500939917812167253300837130397521551187129643} a^{4} + \frac{1121187300348109057732517969798475456914276237538}{3168500939917812167253300837130397521551187129643} a^{3} - \frac{406952970715436982695201381056726604442672822425}{3168500939917812167253300837130397521551187129643} a^{2} + \frac{766691800177930671922133579522120499010625918590}{3168500939917812167253300837130397521551187129643} a + \frac{35586974234573182122076125243299203203531316242}{166763207364095377223857938796336711660588796297}$

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

Class group and class number

$C_{405332}$, which has order $405332$ (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

\(\Q(\sqrt{-203}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.14.7.1$x^{14} - 48778 x^{8} + 594823321 x^{2} - 155248886781$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$