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Properties

Label 10.10.572981288913.1
Degree $10$
Signature $[10, 0]$
Discriminant $3^{5}\cdot 11^{9}$
Root discriminant $14.99$
Ramified primes $3, 11$
Class number $1$
Class group Trivial
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![1, -12, 12, 43, -43, -34, 34, 10, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 10*x^8 + 10*x^7 + 34*x^6 - 34*x^5 - 43*x^4 + 43*x^3 + 12*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^10 - x^9 - 10*x^8 + 10*x^7 + 34*x^6 - 34*x^5 - 43*x^4 + 43*x^3 + 12*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} - 10 x^{8} + 10 x^{7} + 34 x^{6} - 34 x^{5} - 43 x^{4} + 43 x^{3} + 12 x^{2} - 12 x + 1 \)

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:  \(572981288913=3^{5}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.99$
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:  $3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(33=3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{33}(32,·)$, $\chi_{33}(1,·)$, $\chi_{33}(2,·)$, $\chi_{33}(4,·)$, $\chi_{33}(8,·)$, $\chi_{33}(16,·)$, $\chi_{33}(17,·)$, $\chi_{33}(25,·)$, $\chi_{33}(29,·)$, $\chi_{33}(31,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$

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:  $9$
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:  \( a^{7} - 7 a^{5} + a^{4} + 13 a^{3} - 4 a^{2} - 4 a + 1 \),  \( a^{6} - 6 a^{4} + 9 a^{2} - 2 \),  \( a^{9} - 9 a^{7} + 27 a^{5} - 29 a^{3} + 6 a \),  \( a^{9} - 9 a^{7} + 27 a^{5} - 30 a^{3} + 9 a \),  \( a - 1 \),  \( a^{5} - 5 a^{3} + 5 a - 1 \),  \( a^{8} - 8 a^{6} + 20 a^{4} + a^{3} - 16 a^{2} - 3 a + 3 \),  \( a^{2} - 3 \),  \( a^{6} + a^{5} - 6 a^{4} - 4 a^{3} + 9 a^{2} + 2 a - 2 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 245.278866244 \)
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

\(\Q(\sqrt{33}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3_11.2t1.1c1$1$ $ 3 \cdot 11 $ $x^{2} - x - 8$ $C_2$ (as 2T1) $1$ $1$
* 1.11.5t1.1c1$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.3_11.10t1.1c1$1$ $ 3 \cdot 11 $ $x^{10} - x^{9} - 10 x^{8} + 10 x^{7} + 34 x^{6} - 34 x^{5} - 43 x^{4} + 43 x^{3} + 12 x^{2} - 12 x + 1$ $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.1c2$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.3_11.10t1.1c2$1$ $ 3 \cdot 11 $ $x^{10} - x^{9} - 10 x^{8} + 10 x^{7} + 34 x^{6} - 34 x^{5} - 43 x^{4} + 43 x^{3} + 12 x^{2} - 12 x + 1$ $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.1c3$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.3_11.10t1.1c3$1$ $ 3 \cdot 11 $ $x^{10} - x^{9} - 10 x^{8} + 10 x^{7} + 34 x^{6} - 34 x^{5} - 43 x^{4} + 43 x^{3} + 12 x^{2} - 12 x + 1$ $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.1c4$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.3_11.10t1.1c4$1$ $ 3 \cdot 11 $ $x^{10} - x^{9} - 10 x^{8} + 10 x^{7} + 34 x^{6} - 34 x^{5} - 43 x^{4} + 43 x^{3} + 12 x^{2} - 12 x + 1$ $C_{10}$ (as 10T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.