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Properties

Label 10.10.1790566527853125.1
Degree $10$
Signature $[10, 0]$
Discriminant $3^{5}\cdot 5^{5}\cdot 11^{9}$
Root discriminant $33.52$
Ramified primes $3, 5, 11$
Class number $2$
Class group $[2]$
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![-1451, -9813, 9813, 4267, -4267, -661, 661, 43, -43, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 43*x^8 + 43*x^7 + 661*x^6 - 661*x^5 - 4267*x^4 + 4267*x^3 + 9813*x^2 - 9813*x - 1451)
 
gp: K = bnfinit(x^10 - x^9 - 43*x^8 + 43*x^7 + 661*x^6 - 661*x^5 - 4267*x^4 + 4267*x^3 + 9813*x^2 - 9813*x - 1451, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} - 43 x^{8} + 43 x^{7} + 661 x^{6} - 661 x^{5} - 4267 x^{4} + 4267 x^{3} + 9813 x^{2} - 9813 x - 1451 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1790566527853125=3^{5}\cdot 5^{5}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.52$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(164,·)$, $\chi_{165}(134,·)$, $\chi_{165}(136,·)$, $\chi_{165}(74,·)$, $\chi_{165}(16,·)$, $\chi_{165}(149,·)$, $\chi_{165}(91,·)$, $\chi_{165}(29,·)$, $\chi_{165}(31,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{989} a^{6} + \frac{389}{989} a^{5} - \frac{24}{989} a^{4} + \frac{132}{989} a^{3} + \frac{144}{989} a^{2} + \frac{461}{989} a - \frac{128}{989}$, $\frac{1}{989} a^{7} - \frac{28}{989} a^{5} - \frac{422}{989} a^{4} + \frac{224}{989} a^{3} - \frac{171}{989} a^{2} - \frac{448}{989} a + \frac{342}{989}$, $\frac{1}{989} a^{8} - \frac{409}{989} a^{5} - \frac{448}{989} a^{4} - \frac{431}{989} a^{3} - \frac{372}{989} a^{2} + \frac{393}{989} a + \frac{372}{989}$, $\frac{1}{989} a^{9} + \frac{413}{989} a^{5} - \frac{357}{989} a^{4} + \frac{210}{989} a^{3} - \frac{51}{989} a^{2} + \frac{22}{989} a + \frac{65}{989}$

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

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  \( 10474.5233862 \)
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

\(\Q(\sqrt{165}) \), \(\Q(\zeta_{11})^+\)

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 R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$