# Properties

 Label 10.10.572981288913.1 Degree $10$ Signature $[10, 0]$ Discriminant $572981288913$ Root discriminant $$14.99$$ Ramified primes see page Class number $1$ Class group trivial Galois group $C_{10}$ (as 10T1)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 12, 43, -43, -34, 34, 10, -10, -1, 1]);

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $10$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[10, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$572981288913$$ 572981288913 $$\medspace = 3^{5}\cdot 11^{9}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$14.99$$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $$3$$, $$11$$ 3, 11 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Gal(K/\Q) }$: $10$ 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 $$a$$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: Yes Index: $1$ Inessential primes: None

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $9$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ -1  (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $a^{7}-7a^{5}+a^{4}+13a^{3}-4a^{2}-4a+1$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{9}-9a^{7}+27a^{5}-29a^{3}+6a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a-1$, $a^{5}-5a^{3}+5a-1$, $a^{8}-8a^{6}+20a^{4}+a^{3}-16a^{2}-3a+3$, $a^{2}-3$, $a^{6}+a^{5}-6a^{4}-4a^{3}+9a^{2}+2a-2$ 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$245.278866244$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{10}\cdot(2\pi)^{0}\cdot 245.278866244 \cdot 1}{2\sqrt{572981288913}}\approx 0.165905151338$

## Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 10 The 10 conjugacy class representatives for $C_{10}$ Character table for $C_{10}$

## Intermediate fields

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

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

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

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.33.2t1.a.a$1$ $3 \cdot 11$ $$\Q(\sqrt{33})$$ $C_2$ (as 2T1) $1$ $1$
* 1.11.5t1.a.a$1$ $11$ $$\Q(\zeta_{11})^+$$ $C_5$ (as 5T1) $0$ $1$
* 1.33.10t1.b.a$1$ $3 \cdot 11$ $$\Q(\zeta_{33})^+$$ $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.b$1$ $11$ $$\Q(\zeta_{11})^+$$ $C_5$ (as 5T1) $0$ $1$
* 1.33.10t1.b.b$1$ $3 \cdot 11$ $$\Q(\zeta_{33})^+$$ $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.c$1$ $11$ $$\Q(\zeta_{11})^+$$ $C_5$ (as 5T1) $0$ $1$
* 1.33.10t1.b.c$1$ $3 \cdot 11$ $$\Q(\zeta_{33})^+$$ $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.d$1$ $11$ $$\Q(\zeta_{11})^+$$ $C_5$ (as 5T1) $0$ $1$
* 1.33.10t1.b.d$1$ $3 \cdot 11$ $$\Q(\zeta_{33})^+$$ $C_{10}$ (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 *.