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Properties

Label 14.10.2955096450...3361.1
Degree $14$
Signature $[10, 2]$
Discriminant $17^{4}\cdot 29^{12}$
Root discriminant $40.28$
Ramified primes $17, 29$
Class number $1$
Class group Trivial
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![-29, -1479, 2494, 1276, -2204, 754, -841, -550, 1031, -85, -194, 62, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 3*x^12 + 62*x^11 - 194*x^10 - 85*x^9 + 1031*x^8 - 550*x^7 - 841*x^6 + 754*x^5 - 2204*x^4 + 1276*x^3 + 2494*x^2 - 1479*x - 29)
 
gp: K = bnfinit(x^14 - 6*x^13 + 3*x^12 + 62*x^11 - 194*x^10 - 85*x^9 + 1031*x^8 - 550*x^7 - 841*x^6 + 754*x^5 - 2204*x^4 + 1276*x^3 + 2494*x^2 - 1479*x - 29, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 3 x^{12} + 62 x^{11} - 194 x^{10} - 85 x^{9} + 1031 x^{8} - 550 x^{7} - 841 x^{6} + 754 x^{5} - 2204 x^{4} + 1276 x^{3} + 2494 x^{2} - 1479 x - 29 \)

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:  \(29550964508103979773361=17^{4}\cdot 29^{12}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{29} a^{11} + \frac{9}{29} a^{10} + \frac{5}{29} a^{9} - \frac{7}{29} a^{8} - \frac{13}{29} a^{7}$, $\frac{1}{493} a^{12} - \frac{8}{493} a^{11} + \frac{171}{493} a^{10} - \frac{2}{29} a^{9} + \frac{77}{493} a^{8} - \frac{69}{493} a^{7} + \frac{6}{17} a^{6} + \frac{6}{17} a^{5} + \frac{4}{17} a^{4} - \frac{5}{17} a^{3} - \frac{2}{17} a^{2} + \frac{2}{17} a - \frac{1}{17}$, $\frac{1}{1605040775626740281} a^{13} - \frac{1446748764746728}{1605040775626740281} a^{12} + \frac{9849173304027911}{1605040775626740281} a^{11} - \frac{541348254988095341}{1605040775626740281} a^{10} + \frac{402808947529088721}{1605040775626740281} a^{9} - \frac{457608374893894136}{1605040775626740281} a^{8} + \frac{24926902976286484}{1605040775626740281} a^{7} - \frac{20988702877998330}{55346233642301389} a^{6} + \frac{22986649706755566}{55346233642301389} a^{5} + \frac{13428257784823983}{55346233642301389} a^{4} + \frac{2983407122992312}{55346233642301389} a^{3} - \frac{714360590749403}{3255660802488317} a^{2} + \frac{5231018150805665}{55346233642301389} a + \frac{27505452867807527}{55346233642301389}$

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:  \( 1888920.97172 \)
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 ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$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$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$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}$