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Properties

Label 14.0.38011043274...8671.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,13^{7}\cdot 347^{7}$
Root discriminant $67.16$
Ramified primes $13, 347$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1351337, -1106911, 487049, 243329, 101717, 593, 13199, 3707, 2691, -1171, 87, -17, 24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 24*x^12 - 17*x^11 + 87*x^10 - 1171*x^9 + 2691*x^8 + 3707*x^7 + 13199*x^6 + 593*x^5 + 101717*x^4 + 243329*x^3 + 487049*x^2 - 1106911*x + 1351337)
 
gp: K = bnfinit(x^14 - 4*x^13 + 24*x^12 - 17*x^11 + 87*x^10 - 1171*x^9 + 2691*x^8 + 3707*x^7 + 13199*x^6 + 593*x^5 + 101717*x^4 + 243329*x^3 + 487049*x^2 - 1106911*x + 1351337, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 24 x^{12} - 17 x^{11} + 87 x^{10} - 1171 x^{9} + 2691 x^{8} + 3707 x^{7} + 13199 x^{6} + 593 x^{5} + 101717 x^{4} + 243329 x^{3} + 487049 x^{2} - 1106911 x + 1351337 \)

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:  \(-38011043274806994636408671=-\,13^{7}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.16$
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:  $13, 347$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{13} a^{9} + \frac{3}{13} a^{8} + \frac{3}{13} a^{6} - \frac{4}{13} a^{5} - \frac{1}{13} a^{3} - \frac{3}{13} a^{2}$, $\frac{1}{247} a^{10} + \frac{1}{247} a^{9} + \frac{46}{247} a^{8} - \frac{49}{247} a^{7} - \frac{49}{247} a^{6} + \frac{8}{247} a^{5} - \frac{66}{247} a^{4} - \frac{66}{247} a^{3} - \frac{59}{247} a^{2} - \frac{6}{19} a$, $\frac{1}{247} a^{11} + \frac{7}{247} a^{9} + \frac{2}{13} a^{8} - \frac{3}{13} a^{6} + \frac{6}{19} a^{5} + \frac{45}{247} a^{3} + \frac{5}{13} a^{2} + \frac{6}{19} a$, $\frac{1}{26429} a^{12} + \frac{4}{2033} a^{11} + \frac{51}{26429} a^{10} + \frac{693}{26429} a^{9} - \frac{12796}{26429} a^{8} - \frac{8882}{26429} a^{7} - \frac{2325}{26429} a^{6} - \frac{249}{1391} a^{5} + \frac{12455}{26429} a^{4} + \frac{10399}{26429} a^{3} - \frac{1530}{26429} a^{2} - \frac{921}{2033} a + \frac{50}{107}$, $\frac{1}{9609956244181112985624522875201} a^{13} - \frac{165777206245200818290701045}{9609956244181112985624522875201} a^{12} - \frac{11324580882538856731959865160}{9609956244181112985624522875201} a^{11} - \frac{12299073061410085921173147252}{9609956244181112985624522875201} a^{10} - \frac{75194895682912232714551727532}{9609956244181112985624522875201} a^{9} + \frac{25877167127135225647121997682}{9609956244181112985624522875201} a^{8} + \frac{79842753314390529914122651924}{505787170746374367664448572379} a^{7} + \frac{2903592558383948894703266066216}{9609956244181112985624522875201} a^{6} + \frac{4062130740350242282343634247765}{9609956244181112985624522875201} a^{5} + \frac{4626402295262858942005041918231}{9609956244181112985624522875201} a^{4} + \frac{2230704072971178608340820998822}{9609956244181112985624522875201} a^{3} - \frac{2207604196492913525136116041481}{9609956244181112985624522875201} a^{2} - \frac{44240068101762124995589737565}{739227403398547152740347913477} a - \frac{19065622215670036522680319063}{38906705442028797512649890183}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( -1 \) (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:  \( 1766053.09152 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-4511}) \), 7.1.91794884831.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.91794884831.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
347Data not computed