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Properties

Label 14.0.47347025662...3488.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{8}\cdot 3^{15}\cdot 7^{6}\cdot 2221^{3}$
Root discriminant $57.88$
Ramified primes $2, 3, 7, 2221$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T31

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4977, 8415, 31482, 50358, 74250, 44055, 23428, 6670, 754, -19, 17, 29, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 11*x^12 + 29*x^11 + 17*x^10 - 19*x^9 + 754*x^8 + 6670*x^7 + 23428*x^6 + 44055*x^5 + 74250*x^4 + 50358*x^3 + 31482*x^2 + 8415*x + 4977)
 
gp: K = bnfinit(x^14 - 2*x^13 - 11*x^12 + 29*x^11 + 17*x^10 - 19*x^9 + 754*x^8 + 6670*x^7 + 23428*x^6 + 44055*x^5 + 74250*x^4 + 50358*x^3 + 31482*x^2 + 8415*x + 4977, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 11 x^{12} + 29 x^{11} + 17 x^{10} - 19 x^{9} + 754 x^{8} + 6670 x^{7} + 23428 x^{6} + 44055 x^{5} + 74250 x^{4} + 50358 x^{3} + 31482 x^{2} + 8415 x + 4977 \)

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:  \(-4734702566212726100383488=-\,2^{8}\cdot 3^{15}\cdot 7^{6}\cdot 2221^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.88$
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:  $2, 3, 7, 2221$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{2}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a$, $\frac{1}{183682227359598454637632164672} a^{13} - \frac{6104950537667724754057151455}{183682227359598454637632164672} a^{12} - \frac{166097132527633307256224297}{22960278419949806829704020584} a^{11} - \frac{2211403032413973060631729385}{61227409119866151545877388224} a^{10} - \frac{46134157191622264442377583}{2870034802493725853713002573} a^{9} - \frac{58492292542456381334588507027}{183682227359598454637632164672} a^{8} - \frac{7000253773749421001710699981}{61227409119866151545877388224} a^{7} - \frac{68779006176317795334909297415}{183682227359598454637632164672} a^{6} - \frac{35950712234236880426241855601}{183682227359598454637632164672} a^{5} - \frac{10254087999514852331652468263}{45920556839899613659408041168} a^{4} + \frac{4980199714398994816197992617}{30613704559933075772938694112} a^{3} - \frac{410850162755523267176330399}{7653426139983268943234673528} a^{2} - \frac{1564338168971526140598040373}{30613704559933075772938694112} a - \frac{1664498948819179164768516867}{20409136373288717181959129408}$

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:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{23789761311551}{1146512523787342272} a^{13} + \frac{175106167262689}{1146512523787342272} a^{12} - \frac{10962156220129}{143314065473417784} a^{11} - \frac{1768392181581115}{1146512523787342272} a^{10} + \frac{56345682178439}{17914258184177223} a^{9} + \frac{845371202854445}{1146512523787342272} a^{8} - \frac{20682891078922535}{1146512523787342272} a^{7} - \frac{66928910098147271}{1146512523787342272} a^{6} + \frac{245580737115254095}{1146512523787342272} a^{5} + \frac{113469721103679059}{95542710315611856} a^{4} + \frac{385465250555507689}{191085420631223712} a^{3} + \frac{212428520169948649}{47771355157805928} a^{2} + \frac{105815918913964665}{63695140210407904} a + \frac{176751331810978301}{127390280420815808} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 45351348.5741 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T31:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1176
The 17 conjugacy class representatives for [D(7)^2:3_3]2
Character table for [D(7)^2:3_3]2

Intermediate fields

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

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

Sibling fields

Degree 28 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 ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
2221Data not computed