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Properties

Label 10.0.1940336830676403.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 41^{8}$
Root discriminant $33.79$
Ramified primes $3, 41$
Class number $16$
Class group $[2, 2, 2, 2]$
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![81, 189, 486, 183, 370, 131, 230, 26, 17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 17*x^8 + 26*x^7 + 230*x^6 + 131*x^5 + 370*x^4 + 183*x^3 + 486*x^2 + 189*x + 81)
 
gp: K = bnfinit(x^10 - x^9 + 17*x^8 + 26*x^7 + 230*x^6 + 131*x^5 + 370*x^4 + 183*x^3 + 486*x^2 + 189*x + 81, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 17 x^{8} + 26 x^{7} + 230 x^{6} + 131 x^{5} + 370 x^{4} + 183 x^{3} + 486 x^{2} + 189 x + 81 \)

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:  \(-1940336830676403=-\,3^{5}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.79$
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:  $3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(123=3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{123}(1,·)$, $\chi_{123}(98,·)$, $\chi_{123}(100,·)$, $\chi_{123}(37,·)$, $\chi_{123}(10,·)$, $\chi_{123}(16,·)$, $\chi_{123}(83,·)$, $\chi_{123}(119,·)$, $\chi_{123}(59,·)$, $\chi_{123}(92,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{2}{27} a^{6} + \frac{2}{27} a^{5} - \frac{10}{27} a^{4} + \frac{8}{27} a^{3} + \frac{4}{27} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{1636022043} a^{9} - \frac{17556646}{1636022043} a^{8} - \frac{22480090}{1636022043} a^{7} - \frac{270416833}{1636022043} a^{6} - \frac{19226125}{1636022043} a^{5} - \frac{725044636}{1636022043} a^{4} + \frac{745614148}{1636022043} a^{3} + \frac{5543014}{181780227} a^{2} - \frac{12724610}{60593409} a + \frac{4649614}{60593409}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$

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:  \( \frac{2983567}{545340681} a^{9} - \frac{2984416}{545340681} a^{8} + \frac{50334479}{545340681} a^{7} + \frac{76245317}{545340681} a^{6} + \frac{681684926}{545340681} a^{5} + \frac{353606972}{545340681} a^{4} + \frac{977186905}{545340681} a^{3} + \frac{12752341}{60593409} a^{2} + \frac{15664892}{6732601} a + \frac{18254254}{20197803} \) (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:  \( 1973.27972308 \)
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

\(\Q(\sqrt{-3}) \), 5.5.2825761.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$41$41.10.8.1$x^{10} - 27101 x^{5} + 418286592$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$