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Properties

Label 14.0.18875516063...4375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 5^{7}\cdot 73^{7}$
Root discriminant $33.09$
Ramified primes $3, 5, 73$
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![9855, 3285, 15756, -13533, 7159, -4381, 2916, -743, -223, 186, -22, 7, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 7*x^11 - 22*x^10 + 186*x^9 - 223*x^8 - 743*x^7 + 2916*x^6 - 4381*x^5 + 7159*x^4 - 13533*x^3 + 15756*x^2 + 3285*x + 9855)
 
gp: K = bnfinit(x^14 - 4*x^13 + 7*x^11 - 22*x^10 + 186*x^9 - 223*x^8 - 743*x^7 + 2916*x^6 - 4381*x^5 + 7159*x^4 - 13533*x^3 + 15756*x^2 + 3285*x + 9855, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 7 x^{11} - 22 x^{10} + 186 x^{9} - 223 x^{8} - 743 x^{7} + 2916 x^{6} - 4381 x^{5} + 7159 x^{4} - 13533 x^{3} + 15756 x^{2} + 3285 x + 9855 \)

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:  \(-1887551606348838984375=-\,3^{7}\cdot 5^{7}\cdot 73^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.09$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{15} a^{7} + \frac{1}{15} a^{6} + \frac{7}{15} a^{5} + \frac{7}{15} a^{4} + \frac{1}{15} a^{3} + \frac{1}{15} a^{2}$, $\frac{1}{15} a^{8} + \frac{1}{15} a^{6} + \frac{4}{15} a^{4} - \frac{2}{5} a^{2}$, $\frac{1}{225} a^{9} - \frac{2}{75} a^{7} - \frac{3}{25} a^{6} - \frac{11}{25} a^{4} - \frac{88}{225} a^{3} - \frac{8}{25} a^{2} - \frac{1}{15} a + \frac{2}{5}$, $\frac{1}{225} a^{10} - \frac{2}{75} a^{8} + \frac{1}{75} a^{7} + \frac{2}{15} a^{6} + \frac{37}{75} a^{5} - \frac{103}{225} a^{4} - \frac{14}{75} a^{3} + \frac{1}{15} a^{2} + \frac{2}{5} a$, $\frac{1}{225} a^{11} + \frac{1}{75} a^{8} - \frac{2}{75} a^{7} + \frac{8}{75} a^{6} - \frac{103}{225} a^{5} - \frac{37}{75} a^{4} - \frac{7}{25} a^{3} - \frac{14}{75} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{3375} a^{12} - \frac{1}{1125} a^{11} - \frac{1}{3375} a^{10} - \frac{11}{375} a^{8} + \frac{4}{1125} a^{7} + \frac{41}{3375} a^{6} - \frac{151}{1125} a^{5} - \frac{211}{675} a^{4} - \frac{389}{1125} a^{3} - \frac{406}{1125} a^{2} - \frac{13}{75} a + \frac{1}{25}$, $\frac{1}{5396573768326875} a^{13} + \frac{178989733763}{5396573768326875} a^{12} + \frac{8058135026996}{5396573768326875} a^{11} + \frac{6162039880564}{5396573768326875} a^{10} + \frac{2449719823022}{1798857922775625} a^{9} - \frac{2486479929164}{1798857922775625} a^{8} + \frac{70947160328513}{5396573768326875} a^{7} + \frac{856641931610578}{5396573768326875} a^{6} - \frac{1116330944533583}{5396573768326875} a^{5} + \frac{1850018238196058}{5396573768326875} a^{4} + \frac{89750218330063}{359771584555125} a^{3} - \frac{321909141867131}{1798857922775625} a^{2} - \frac{3977610369008}{23984772303675} a - \frac{13720391364019}{39974620506125}$

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:  \( 560331.922424 \) (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

\(\Q(\sqrt{-1095}) \), 7.1.1312932375.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.1312932375.1

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 ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$73$73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$