↑

Properties

Label 14.14.4024886456...1472.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{21}\cdot 3^{16}\cdot 13^{8}\cdot 2719^{4}$
Root discriminant $411.74$
Ramified primes $2, 3, 13, 2719$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T26

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5730505137, 7815136932, 223044192, -2641431330, -109694286, 348974586, 53650141, -11384888, -2593249, 110038, 45973, 146, -346, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 346*x^12 + 146*x^11 + 45973*x^10 + 110038*x^9 - 2593249*x^8 - 11384888*x^7 + 53650141*x^6 + 348974586*x^5 - 109694286*x^4 - 2641431330*x^3 + 223044192*x^2 + 7815136932*x - 5730505137)
 
gp: K = bnfinit(x^14 - 4*x^13 - 346*x^12 + 146*x^11 + 45973*x^10 + 110038*x^9 - 2593249*x^8 - 11384888*x^7 + 53650141*x^6 + 348974586*x^5 - 109694286*x^4 - 2641431330*x^3 + 223044192*x^2 + 7815136932*x - 5730505137, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 346 x^{12} + 146 x^{11} + 45973 x^{10} + 110038 x^{9} - 2593249 x^{8} - 11384888 x^{7} + 53650141 x^{6} + 348974586 x^{5} - 109694286 x^{4} - 2641431330 x^{3} + 223044192 x^{2} + 7815136932 x - 5730505137 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4024886456410352267069803892909801472=2^{21}\cdot 3^{16}\cdot 13^{8}\cdot 2719^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $411.74$
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:  $2, 3, 13, 2719$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{4}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{4}{9} a^{3}$, $\frac{1}{93895283111766585150353382525549008534749389} a^{13} - \frac{797251438403991296773136981203225225493764}{93895283111766585150353382525549008534749389} a^{12} + \frac{1293556054080945168118482625539878279606351}{93895283111766585150353382525549008534749389} a^{11} - \frac{11182061025168152074024566123450653568503}{156753394176571928464696798874038411577211} a^{10} + \frac{7713943992721071223517053503284313164610763}{93895283111766585150353382525549008534749389} a^{9} - \frac{7347480876580988473897805372375647462057148}{93895283111766585150353382525549008534749389} a^{8} + \frac{38141191967672129721824701168142992807964177}{93895283111766585150353382525549008534749389} a^{7} + \frac{35182261565781373479492847068609803166869830}{93895283111766585150353382525549008534749389} a^{6} + \frac{29318202124088884395219125459030095674916648}{93895283111766585150353382525549008534749389} a^{5} + \frac{4068090984953747438524354991076827488090094}{10432809234640731683372598058394334281638821} a^{4} + \frac{15492308658377471956267472593648790998039718}{31298427703922195050117794175183002844916463} a^{3} + \frac{364938323067623312730596296683255647727194}{3477603078213577227790866019464778093879607} a^{2} - \frac{1230367647969058280979523870559567559410111}{3477603078213577227790866019464778093879607} a + \frac{276513173425669285647014934218977071496604}{3477603078213577227790866019464778093879607}$

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:  $13$
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:  \( 106512528185000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T26:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 882
The 20 conjugacy class representatives for 1/2[1/2.F_42(7)^2]2
Character table for 1/2[1/2.F_42(7)^2]2

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Sibling fields

Degree 21 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.14.21.34$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$2$$7$$21$$C_{14}$$[3]^{7}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
2719Data not computed