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Properties

Label 10.0.889269299311.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,17^{2}\cdot 79^{5}$
Root discriminant $15.66$
Ramified primes $17, 79$
Class number $1$
Class group Trivial
Galois group $(C_2^4 : C_5) : C_2$ (as 10T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63, -195, 316, -350, 299, -193, 107, -44, 17, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 17*x^8 - 44*x^7 + 107*x^6 - 193*x^5 + 299*x^4 - 350*x^3 + 316*x^2 - 195*x + 63)
 
gp: K = bnfinit(x^10 - 4*x^9 + 17*x^8 - 44*x^7 + 107*x^6 - 193*x^5 + 299*x^4 - 350*x^3 + 316*x^2 - 195*x + 63, 1)
 

Normalized defining polynomial

\( x^{10} - 4 x^{9} + 17 x^{8} - 44 x^{7} + 107 x^{6} - 193 x^{5} + 299 x^{4} - 350 x^{3} + 316 x^{2} - 195 x + 63 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-889269299311=-\,17^{2}\cdot 79^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.66$
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:  $17, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a - \frac{1}{3}$

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:  $4$
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:  \( 86.0691002849 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 10T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $(C_2^4 : C_5) : C_2$
Character table for $(C_2^4 : C_5) : C_2$

Intermediate fields

5.1.6241.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.4.3.1$x^{4} + 158$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$