↑

Properties

Label 10.0.207252522098163.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 31^{8}$
Root discriminant $27.02$
Ramified primes $3, 31$
Class number $5$
Class group $[5]$
Galois group $C_{10}$ (as 10T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -5, 106, -99, 448, -255, 164, -30, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 13*x^8 - 30*x^7 + 164*x^6 - 255*x^5 + 448*x^4 - 99*x^3 + 106*x^2 - 5*x + 25)
 
gp: K = bnfinit(x^10 - x^9 + 13*x^8 - 30*x^7 + 164*x^6 - 255*x^5 + 448*x^4 - 99*x^3 + 106*x^2 - 5*x + 25, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 13 x^{8} - 30 x^{7} + 164 x^{6} - 255 x^{5} + 448 x^{4} - 99 x^{3} + 106 x^{2} - 5 x + 25 \)

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:  \(-207252522098163=-\,3^{5}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.02$
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:  $3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(93=3\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{93}(32,·)$, $\chi_{93}(1,·)$, $\chi_{93}(2,·)$, $\chi_{93}(35,·)$, $\chi_{93}(4,·)$, $\chi_{93}(70,·)$, $\chi_{93}(8,·)$, $\chi_{93}(64,·)$, $\chi_{93}(47,·)$, $\chi_{93}(16,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{414492635} a^{9} - \frac{26580558}{414492635} a^{8} - \frac{7346259}{82898527} a^{7} + \frac{132602042}{414492635} a^{6} - \frac{17287581}{82898527} a^{5} - \frac{163967618}{414492635} a^{4} - \frac{63821001}{414492635} a^{3} + \frac{20015554}{82898527} a^{2} - \frac{32214086}{82898527} a + \frac{13310823}{82898527}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( \frac{819337}{414492635} a^{9} + \frac{3175178}{414492635} a^{8} + \frac{6393611}{414492635} a^{7} + \frac{5496161}{82898527} a^{6} + \frac{12919128}{414492635} a^{5} + \frac{90165976}{82898527} a^{4} - \frac{672026439}{414492635} a^{3} + \frac{1740696047}{414492635} a^{2} - \frac{155456098}{414492635} a + \frac{85369285}{82898527} \) (order $6$)
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:  \( 485.913224212 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}$ (as 10T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 10
The 10 conjugacy class representatives for $C_{10}$
Character table for $C_{10}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.923521.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ R ${\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$31$31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$