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Properties

Label 10.0.32721738523...0064.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{15}\cdot 3^{5}\cdot 11^{8}\cdot 61^{8}$
Root discriminant $894.30$
Ramified primes $2, 3, 11, 61$
Class number $1420000$ (GRH)
Class group $[2, 10, 10, 10, 710]$ (GRH)
Galois group $C_{10}$ (as 10T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![225499504, -47220384, 37976784, -666304, -110392, 557872, 76072, -1552, -505, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 505*x^8 - 1552*x^7 + 76072*x^6 + 557872*x^5 - 110392*x^4 - 666304*x^3 + 37976784*x^2 - 47220384*x + 225499504)
 
gp: K = bnfinit(x^10 - 2*x^9 - 505*x^8 - 1552*x^7 + 76072*x^6 + 557872*x^5 - 110392*x^4 - 666304*x^3 + 37976784*x^2 - 47220384*x + 225499504, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} - 505 x^{8} - 1552 x^{7} + 76072 x^{6} + 557872 x^{5} - 110392 x^{4} - 666304 x^{3} + 37976784 x^{2} - 47220384 x + 225499504 \)

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:  \(-327217385234372376075381080064=-\,2^{15}\cdot 3^{5}\cdot 11^{8}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $894.30$
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:  $2, 3, 11, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(16104=2^{3}\cdot 3\cdot 11\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{16104}(1,·)$, $\chi_{16104}(6341,·)$, $\chi_{16104}(7781,·)$, $\chi_{16104}(9025,·)$, $\chi_{16104}(10013,·)$, $\chi_{16104}(10085,·)$, $\chi_{16104}(10465,·)$, $\chi_{16104}(12697,·)$, $\chi_{16104}(12769,·)$, $\chi_{16104}(13421,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{6} + \frac{1}{16} a^{5} - \frac{1}{32} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{64} a^{7} - \frac{5}{64} a^{5} + \frac{3}{32} a^{4} - \frac{3}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{16} a - \frac{1}{8}$, $\frac{1}{27392} a^{8} - \frac{85}{13696} a^{7} - \frac{407}{27392} a^{6} + \frac{549}{6848} a^{5} + \frac{1125}{13696} a^{4} - \frac{1591}{3424} a^{3} + \frac{1831}{6848} a^{2} + \frac{759}{1712} a + \frac{11}{32}$, $\frac{1}{99297532890098804566528} a^{9} + \frac{42631196552134079}{3103047902815587642704} a^{8} + \frac{39315619638465950445}{99297532890098804566528} a^{7} - \frac{532915051439035904601}{49648766445049402283264} a^{6} - \frac{5500481152427190098555}{49648766445049402283264} a^{5} - \frac{2290936043427313124833}{24824383222524701141632} a^{4} - \frac{7823976281736782319613}{24824383222524701141632} a^{3} - \frac{1815838026869596197207}{12412191611262350570816} a^{2} - \frac{6068226185128749263111}{12412191611262350570816} a + \frac{3771930673437445507}{58000895379730610144}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{10}\times C_{10}\times C_{710}$, which has order $1420000$ (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:  $4$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 400186.6439256542 \) (assuming GRH)
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

\(\Q(\sqrt{-6}) \), 5.5.202716958081.1

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 R R ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$61$61.10.8.3$x^{10} + 183 x^{5} + 14884$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$