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Properties

Label 14.2.29550964508...3361.1
Degree $14$
Signature $[2, 6]$
Discriminant $17^{4}\cdot 29^{12}$
Root discriminant $40.28$
Ramified primes $17, 29$
Class number $4$
Class group $[2, 2]$
Galois group $C_2^3:F_8$ (as 14T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-173, -492, -917, 920, -162, -312, 287, -238, 131, -86, 66, -41, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 19*x^12 - 41*x^11 + 66*x^10 - 86*x^9 + 131*x^8 - 238*x^7 + 287*x^6 - 312*x^5 - 162*x^4 + 920*x^3 - 917*x^2 - 492*x - 173)
 
gp: K = bnfinit(x^14 - 6*x^13 + 19*x^12 - 41*x^11 + 66*x^10 - 86*x^9 + 131*x^8 - 238*x^7 + 287*x^6 - 312*x^5 - 162*x^4 + 920*x^3 - 917*x^2 - 492*x - 173, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 19 x^{12} - 41 x^{11} + 66 x^{10} - 86 x^{9} + 131 x^{8} - 238 x^{7} + 287 x^{6} - 312 x^{5} - 162 x^{4} + 920 x^{3} - 917 x^{2} - 492 x - 173 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(29550964508103979773361=17^{4}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.28$
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:  $17, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{59} a^{12} - \frac{28}{59} a^{11} + \frac{10}{59} a^{10} + \frac{11}{59} a^{9} + \frac{5}{59} a^{8} + \frac{9}{59} a^{7} - \frac{6}{59} a^{6} - \frac{8}{59} a^{5} + \frac{24}{59} a^{4} - \frac{29}{59} a^{3} - \frac{10}{59} a^{2} - \frac{28}{59} a - \frac{10}{59}$, $\frac{1}{40696410982317659} a^{13} - \frac{17550519235984}{40696410982317659} a^{12} - \frac{17048015950120714}{40696410982317659} a^{11} + \frac{13124543638840454}{40696410982317659} a^{10} + \frac{8869201175324414}{40696410982317659} a^{9} - \frac{16467889585645315}{40696410982317659} a^{8} + \frac{10648071349104967}{40696410982317659} a^{7} - \frac{3097445362246286}{40696410982317659} a^{6} + \frac{5538094913290969}{40696410982317659} a^{5} - \frac{7083256869227657}{40696410982317659} a^{4} - \frac{9073813384566490}{40696410982317659} a^{3} + \frac{617774595920978}{2393906528371627} a^{2} + \frac{5159802372622468}{40696410982317659} a - \frac{4973146117639970}{40696410982317659}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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:  \( 109772.145402 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:F_8$ (as 14T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 448
The 16 conjugacy class representatives for $C_2^3:F_8$
Character table for $C_2^3:F_8$

Intermediate fields

7.7.594823321.1

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$