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Properties

Label 14.2.47178111989...9241.1
Degree $14$
Signature $[2, 6]$
Discriminant $17^{5}\cdot 23^{3}\cdot 64879^{3}$
Root discriminant $57.86$
Ramified primes $17, 23, 64879$
Class number $64$ (GRH)
Class group $[4, 16]$ (GRH)
Galois group 14T54

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![318848, -1853568, 191744, -841056, 329208, -204216, 117192, -35611, 18994, -3926, 1616, -227, 67, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 67*x^12 - 227*x^11 + 1616*x^10 - 3926*x^9 + 18994*x^8 - 35611*x^7 + 117192*x^6 - 204216*x^5 + 329208*x^4 - 841056*x^3 + 191744*x^2 - 1853568*x + 318848)
 
gp: K = bnfinit(x^14 - 5*x^13 + 67*x^12 - 227*x^11 + 1616*x^10 - 3926*x^9 + 18994*x^8 - 35611*x^7 + 117192*x^6 - 204216*x^5 + 329208*x^4 - 841056*x^3 + 191744*x^2 - 1853568*x + 318848, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 67 x^{12} - 227 x^{11} + 1616 x^{10} - 3926 x^{9} + 18994 x^{8} - 35611 x^{7} + 117192 x^{6} - 204216 x^{5} + 329208 x^{4} - 841056 x^{3} + 191744 x^{2} - 1853568 x + 318848 \)

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:  \(4717811198979405309189241=17^{5}\cdot 23^{3}\cdot 64879^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.86$
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:  $17, 23, 64879$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} + \frac{1}{32} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{7}{16} a^{6} - \frac{3}{32} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{13277902680971731541354257465792} a^{13} - \frac{73782756553743716134045849715}{13277902680971731541354257465792} a^{12} - \frac{4217152416762770105328472143}{13277902680971731541354257465792} a^{11} + \frac{40495055571921020176818935315}{13277902680971731541354257465792} a^{10} - \frac{372862163329971337059397215427}{6638951340485865770677128732896} a^{9} + \frac{1544067824891837477080337717453}{6638951340485865770677128732896} a^{8} + \frac{964968216798589352892799752751}{6638951340485865770677128732896} a^{7} + \frac{3017536295979849662620601790601}{13277902680971731541354257465792} a^{6} - \frac{3312651758920590211442850130687}{6638951340485865770677128732896} a^{5} - \frac{155753172383567118498182692585}{829868917560733221334641091612} a^{4} - \frac{185395353745995430833166065701}{829868917560733221334641091612} a^{3} - \frac{3826328843997536714069456463}{414934458780366610667320545806} a^{2} + \frac{46089589002062075255689192911}{207467229390183305333660272903} a - \frac{22008632530004465537746033106}{207467229390183305333660272903}$

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

Class group and class number

$C_{4}\times C_{16}$, which has order $64$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 54512.7529812 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T54:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 322560
The 55 conjugacy class representatives for 1/2[2^7]S(7) are not computed
Character table for 1/2[2^7]S(7) is not computed

Intermediate fields

7.7.25367689.1

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 28 sibling: 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
64879Data not computed