↑

Properties

Label 14.0.10405264994...4027.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{19}\cdot 547^{4}$
Root discriminant $26.90$
Ramified primes $3, 547$
Class number $4$
Class group $[2, 2]$
Galois group $A_7\times C_2$ (as 14T47)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, 185, -473, -191, 734, -152, -394, 223, 23, -46, 22, -19, 10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 10*x^12 - 19*x^11 + 22*x^10 - 46*x^9 + 23*x^8 + 223*x^7 - 394*x^6 - 152*x^5 + 734*x^4 - 191*x^3 - 473*x^2 + 185*x + 169)
 
gp: K = bnfinit(x^14 - x^13 + 10*x^12 - 19*x^11 + 22*x^10 - 46*x^9 + 23*x^8 + 223*x^7 - 394*x^6 - 152*x^5 + 734*x^4 - 191*x^3 - 473*x^2 + 185*x + 169, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 10 x^{12} - 19 x^{11} + 22 x^{10} - 46 x^{9} + 23 x^{8} + 223 x^{7} - 394 x^{6} - 152 x^{5} + 734 x^{4} - 191 x^{3} - 473 x^{2} + 185 x + 169 \)

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:  \(-104052649942678734027=-\,3^{19}\cdot 547^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.90$
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:  $3, 547$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{93047389823980187} a^{13} - \frac{33942761296374586}{93047389823980187} a^{12} + \frac{32213313696589759}{93047389823980187} a^{11} - \frac{37969123434106820}{93047389823980187} a^{10} - \frac{1996197675394141}{93047389823980187} a^{9} - \frac{36474780937126345}{93047389823980187} a^{8} + \frac{42070990073649010}{93047389823980187} a^{7} + \frac{20176195598284019}{93047389823980187} a^{6} - \frac{7782138712167113}{93047389823980187} a^{5} + \frac{30666035768733132}{93047389823980187} a^{4} + \frac{5983732518634316}{93047389823980187} a^{3} + \frac{6370352879027720}{93047389823980187} a^{2} - \frac{39983039464132224}{93047389823980187} a - \frac{8539810503820304}{93047389823980187}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{870432031}{77272065779} a^{13} - \frac{1268974872}{77272065779} a^{12} + \frac{9176108091}{77272065779} a^{11} - \frac{21441193099}{77272065779} a^{10} + \frac{27490593384}{77272065779} a^{9} - \frac{59261715996}{77272065779} a^{8} + \frac{48536021240}{77272065779} a^{7} + \frac{164093826959}{77272065779} a^{6} - \frac{389886449088}{77272065779} a^{5} + \frac{34544061775}{77272065779} a^{4} + \frac{525770854031}{77272065779} a^{3} - \frac{340335303933}{77272065779} a^{2} - \frac{123765750524}{77272065779} a + \frac{177549653888}{77272065779} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18139.3317602 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_7\times C_2$ (as 14T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5040
The 18 conjugacy class representatives for $A_7\times C_2$
Character table for $A_7\times C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 7.7.1963110249.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.12.18.64$x^{12} + 12 x^{10} - 12 x^{9} - 9 x^{8} - 9 x^{7} - 9 x^{6} - 9 x^{5} - 9 x^{4} + 9 x^{3} + 9$$6$$2$$18$12T40$[2, 2]_{2}^{4}$
547Data not computed