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Properties

Label 14.0.75573382356...4375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 5^{7}\cdot 89^{7}$
Root discriminant $36.54$
Ramified primes $3, 5, 89$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, -729, 5454, 10557, 9963, -6741, 5418, -1569, 187, 104, 2, 17, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 4*x^12 + 17*x^11 + 2*x^10 + 104*x^9 + 187*x^8 - 1569*x^7 + 5418*x^6 - 6741*x^5 + 9963*x^4 + 10557*x^3 + 5454*x^2 - 729*x + 729)
 
gp: K = bnfinit(x^14 - 4*x^13 - 4*x^12 + 17*x^11 + 2*x^10 + 104*x^9 + 187*x^8 - 1569*x^7 + 5418*x^6 - 6741*x^5 + 9963*x^4 + 10557*x^3 + 5454*x^2 - 729*x + 729, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 4 x^{12} + 17 x^{11} + 2 x^{10} + 104 x^{9} + 187 x^{8} - 1569 x^{7} + 5418 x^{6} - 6741 x^{5} + 9963 x^{4} + 10557 x^{3} + 5454 x^{2} - 729 x + 729 \)

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:  \(-7557338235665775234375=-\,3^{7}\cdot 5^{7}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.54$
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:  $3, 5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{45} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{2}{45} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a - \frac{1}{5}$, $\frac{1}{45} a^{9} - \frac{4}{15} a^{5} + \frac{1}{9} a^{3} + \frac{2}{15} a$, $\frac{1}{45} a^{10} + \frac{1}{15} a^{6} + \frac{4}{9} a^{4} + \frac{7}{15} a^{2}$, $\frac{1}{675} a^{11} - \frac{7}{675} a^{10} - \frac{4}{675} a^{9} - \frac{7}{675} a^{8} - \frac{112}{675} a^{7} + \frac{44}{675} a^{6} + \frac{163}{675} a^{5} - \frac{62}{225} a^{4} - \frac{8}{225} a^{3} - \frac{13}{75} a^{2} + \frac{3}{25} a - \frac{6}{25}$, $\frac{1}{2025} a^{12} - \frac{1}{2025} a^{11} - \frac{1}{2025} a^{10} - \frac{1}{2025} a^{9} + \frac{11}{2025} a^{8} + \frac{122}{2025} a^{7} - \frac{38}{2025} a^{6} + \frac{319}{675} a^{5} + \frac{22}{135} a^{4} - \frac{79}{225} a^{3} - \frac{89}{225} a^{2} + \frac{9}{25} a + \frac{3}{25}$, $\frac{1}{1200186000951852375} a^{13} + \frac{113541270341288}{1200186000951852375} a^{12} - \frac{113977865315518}{1200186000951852375} a^{11} + \frac{2559583112081186}{1200186000951852375} a^{10} - \frac{485623745995531}{1200186000951852375} a^{9} - \frac{11008325770111648}{1200186000951852375} a^{8} - \frac{189309443029888994}{1200186000951852375} a^{7} - \frac{16480202365669613}{133354000105761375} a^{6} - \frac{86130211064852002}{400062000317284125} a^{5} - \frac{7153974984474203}{14817111122862375} a^{4} - \frac{57944630359398812}{133354000105761375} a^{3} + \frac{21981067076899973}{44451333368587125} a^{2} - \frac{1410762822705513}{4939037040954125} a - \frac{654112633138874}{4939037040954125}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 1166182.92118 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-1335}) \), 7.1.2379270375.1 x7

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 7 sibling: 7.1.2379270375.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$