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Properties

Label 14.14.6517333038...6048.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{12}\cdot 73^{8}\cdot 1973$
Root discriminant $36.15$
Ramified primes $2, 73, 1973$
Class number $1$
Class group Trivial
Galois group 14T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 7, 38, -261, 325, 320, -749, 43, 509, -176, -121, 67, 4, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 4*x^12 + 67*x^11 - 121*x^10 - 176*x^9 + 509*x^8 + 43*x^7 - 749*x^6 + 320*x^5 + 325*x^4 - 261*x^3 + 38*x^2 + 7*x - 1)
 
gp: K = bnfinit(x^14 - 7*x^13 + 4*x^12 + 67*x^11 - 121*x^10 - 176*x^9 + 509*x^8 + 43*x^7 - 749*x^6 + 320*x^5 + 325*x^4 - 261*x^3 + 38*x^2 + 7*x - 1, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 4 x^{12} + 67 x^{11} - 121 x^{10} - 176 x^{9} + 509 x^{8} + 43 x^{7} - 749 x^{6} + 320 x^{5} + 325 x^{4} - 261 x^{3} + 38 x^{2} + 7 x - 1 \)

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:  \(6517333038313561346048=2^{12}\cdot 73^{8}\cdot 1973\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.15$
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:  $2, 73, 1973$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1031} a^{12} - \frac{6}{1031} a^{11} - \frac{388}{1031} a^{10} - \frac{67}{1031} a^{9} + \frac{85}{1031} a^{8} - \frac{4}{1031} a^{7} - \frac{344}{1031} a^{6} + \frac{212}{1031} a^{5} + \frac{279}{1031} a^{4} + \frac{216}{1031} a^{3} + \frac{71}{1031} a^{2} - \frac{55}{1031} a + \frac{414}{1031}$, $\frac{1}{1031} a^{13} - \frac{424}{1031} a^{11} - \frac{333}{1031} a^{10} - \frac{317}{1031} a^{9} + \frac{506}{1031} a^{8} - \frac{368}{1031} a^{7} + \frac{210}{1031} a^{6} - \frac{511}{1031} a^{5} - \frac{172}{1031} a^{4} + \frac{336}{1031} a^{3} + \frac{371}{1031} a^{2} + \frac{84}{1031} a + \frac{422}{1031}$

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:  $13$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2554485.78647 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2688
The 32 conjugacy class representatives for [2^7]F_21(7)=2wrF_21(7)
Character table for [2^7]F_21(7)=2wrF_21(7) is not computed

Intermediate fields

7.7.1817487424.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.14.12.1$x^{14} - 2 x^{7} + 4$$7$$2$$12$$(C_7:C_3) \times C_2$$[\ ]_{7}^{6}$
73Data not computed
1973Data not computed