↑

Properties

Label 14.2.86965122767...0000.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{8}\cdot 3^{16}\cdot 5^{10}\cdot 7^{24}\cdot 59^{6}$
Root discriminant $2656.07$
Ramified primes $2, 3, 5, 7, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T22

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![91264986397629, 7429846568445, 17926524826973, 4638317668005, -335941176396, -117366985106, 2085856500, 859373823, -17922660, -708246, 118671, -9555, -63, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 63*x^12 - 9555*x^11 + 118671*x^10 - 708246*x^9 - 17922660*x^8 + 859373823*x^7 + 2085856500*x^6 - 117366985106*x^5 - 335941176396*x^4 + 4638317668005*x^3 + 17926524826973*x^2 + 7429846568445*x + 91264986397629)
 
gp: K = bnfinit(x^14 - 63*x^12 - 9555*x^11 + 118671*x^10 - 708246*x^9 - 17922660*x^8 + 859373823*x^7 + 2085856500*x^6 - 117366985106*x^5 - 335941176396*x^4 + 4638317668005*x^3 + 17926524826973*x^2 + 7429846568445*x + 91264986397629, 1)
 

Normalized defining polynomial

\( x^{14} - 63 x^{12} - 9555 x^{11} + 118671 x^{10} - 708246 x^{9} - 17922660 x^{8} + 859373823 x^{7} + 2085856500 x^{6} - 117366985106 x^{5} - 335941176396 x^{4} + 4638317668005 x^{3} + 17926524826973 x^{2} + 7429846568445 x + 91264986397629 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(869651227673543607965251034000845898902500000000=2^{8}\cdot 3^{16}\cdot 5^{10}\cdot 7^{24}\cdot 59^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2656.07$
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:  $2, 3, 5, 7, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{25} a^{8} - \frac{2}{25} a^{7} + \frac{1}{25} a^{6} - \frac{2}{25} a^{5} + \frac{2}{25} a^{3} + \frac{6}{25} a^{2} - \frac{3}{25} a - \frac{4}{25}$, $\frac{1}{75} a^{9} - \frac{1}{25} a^{7} + \frac{1}{15} a^{6} + \frac{16}{75} a^{5} - \frac{6}{25} a^{4} + \frac{1}{3} a^{3} + \frac{4}{75} a^{2} + \frac{2}{15} a + \frac{4}{25}$, $\frac{1}{375} a^{10} + \frac{2}{375} a^{9} - \frac{1}{125} a^{8} - \frac{31}{375} a^{7} - \frac{4}{375} a^{6} - \frac{181}{375} a^{5} + \frac{139}{375} a^{4} + \frac{43}{125} a^{3} - \frac{39}{125} a^{2} - \frac{28}{375} a - \frac{47}{125}$, $\frac{1}{375} a^{11} - \frac{2}{375} a^{9} + \frac{1}{75} a^{8} - \frac{17}{375} a^{7} + \frac{32}{375} a^{6} - \frac{4}{375} a^{5} - \frac{89}{375} a^{4} - \frac{23}{75} a^{3} - \frac{119}{375} a^{2} + \frac{4}{15} a + \frac{24}{125}$, $\frac{1}{1875} a^{12} + \frac{1}{1875} a^{11} - \frac{1}{1875} a^{10} + \frac{1}{375} a^{9} + \frac{1}{125} a^{8} - \frac{1}{1875} a^{7} + \frac{18}{625} a^{6} + \frac{791}{1875} a^{5} + \frac{122}{375} a^{4} + \frac{7}{125} a^{3} - \frac{481}{1875} a^{2} + \frac{93}{625} a + \frac{162}{625}$, $\frac{1}{234898238096534679822440775066866021285729703515922573715965454612786875} a^{13} - \frac{9993754774211978847169335276287020242264819300989918991375516961331}{78299412698844893274146925022288673761909901171974191238655151537595625} a^{12} + \frac{47575017912185813099939649775624503414081754499019637982528132566726}{46979647619306935964488155013373204257145940703184514743193090922557375} a^{11} - \frac{179956381195886291599382627888508350478822165848266336144926835816976}{234898238096534679822440775066866021285729703515922573715965454612786875} a^{10} - \frac{140761764187226306798884230287023209221932995535020263241461579787806}{46979647619306935964488155013373204257145940703184514743193090922557375} a^{9} - \frac{146634886281267487919880780817247458654384681167340020378709761243836}{234898238096534679822440775066866021285729703515922573715965454612786875} a^{8} - \frac{650753664209988462732206148893478606815436935712001570753662023871902}{234898238096534679822440775066866021285729703515922573715965454612786875} a^{7} - \frac{2414339634959327622536955290826298015253402967205433805819540791443217}{46979647619306935964488155013373204257145940703184514743193090922557375} a^{6} - \frac{79805255722008645327225612713182749584790175087998036769333595138245469}{234898238096534679822440775066866021285729703515922573715965454612786875} a^{5} + \frac{3249459251379810058894541620121621610476820370119188211805110496829648}{46979647619306935964488155013373204257145940703184514743193090922557375} a^{4} + \frac{46557129524116651240435011998433159504654742066255621648633786238499674}{234898238096534679822440775066866021285729703515922573715965454612786875} a^{3} + \frac{26387927066865851410479606251072994501331075081486214381753606759551581}{78299412698844893274146925022288673761909901171974191238655151537595625} a^{2} - \frac{2583418474782054465936871605370038693975009594722214620275608199916773}{46979647619306935964488155013373204257145940703184514743193090922557375} a - \frac{18349020425247369086084667363817070770538932199420921941504036490791153}{78299412698844893274146925022288673761909901171974191238655151537595625}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $7$
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:  \( 6801116114790000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T22:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 588
The 10 conjugacy class representatives for [1/6_-.F_42(7)^2]2_2
Character table for [1/6_-.F_42(7)^2]2_2

Intermediate fields

\(\Q(\sqrt{5}) \)

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

Sibling fields

Degree 28 sibling: 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 R R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ R

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.12.16.30$x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324$$3$$4$$16$$C_3 : C_4$$[2]^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
7Data not computed
59Data not computed