↑

Properties

Label 14.10.1126648563...1681.1
Degree $14$
Signature $[10, 2]$
Discriminant $3^{6}\cdot 7^{8}\cdot 173^{6}$
Root discriminant $44.32$
Ramified primes $3, 7, 173$
Class number $1$
Class group Trivial
Galois group 14T41

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -252, -491, 1302, 2662, 16, -1710, -1066, 318, 289, 48, -19, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 17*x^12 - 19*x^11 + 48*x^10 + 289*x^9 + 318*x^8 - 1066*x^7 - 1710*x^6 + 16*x^5 + 2662*x^4 + 1302*x^3 - 491*x^2 - 252*x - 9)
 
gp: K = bnfinit(x^14 - 17*x^12 - 19*x^11 + 48*x^10 + 289*x^9 + 318*x^8 - 1066*x^7 - 1710*x^6 + 16*x^5 + 2662*x^4 + 1302*x^3 - 491*x^2 - 252*x - 9, 1)
 

Normalized defining polynomial

\( x^{14} - 17 x^{12} - 19 x^{11} + 48 x^{10} + 289 x^{9} + 318 x^{8} - 1066 x^{7} - 1710 x^{6} + 16 x^{5} + 2662 x^{4} + 1302 x^{3} - 491 x^{2} - 252 x - 9 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(112664856324815819481681=3^{6}\cdot 7^{8}\cdot 173^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.32$
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:  $3, 7, 173$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{3270161620533070704} a^{13} + \frac{60673820892438519}{1090053873511023568} a^{12} - \frac{15409688395609183}{408770202566633838} a^{11} + \frac{295410289780054997}{3270161620533070704} a^{10} + \frac{52131765906790341}{155721981930146224} a^{9} - \frac{733021189076600737}{1635080810266535352} a^{8} + \frac{69275598879140363}{272513468377755892} a^{7} + \frac{491430693977016709}{1635080810266535352} a^{6} - \frac{67074593572274997}{272513468377755892} a^{5} - \frac{25896377173801607}{817540405133267676} a^{4} - \frac{330736215727464811}{1635080810266535352} a^{3} - \frac{8608375938038111}{19465247741268278} a^{2} + \frac{1060893598707466093}{3270161620533070704} a - \frac{456034886254762377}{1090053873511023568}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12761165.1007 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T41:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2688
The 20 conjugacy class representatives for [2^6]F_42(7)
Character table for [2^6]F_42(7)

Intermediate fields

7.7.12431698517.1

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

Sibling fields

Degree 14 sibling: data not computed
Degree 16 sibling: data not computed
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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$173$$\Q_{173}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{173}$$x + 2$$1$$1$$0$Trivial$[\ ]$
173.12.6.1$x^{12} + 196753246 x^{6} - 154963892093 x^{2} + 9677959952884129$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$