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Properties

Label 10.0.26439622160671.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,31^{9}$
Root discriminant $21.99$
Ramified prime $31$
Class number $3$
Class group $[3]$
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![25, -20, 63, -6, 43, 11, -9, 16, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 2*x^8 + 16*x^7 - 9*x^6 + 11*x^5 + 43*x^4 - 6*x^3 + 63*x^2 - 20*x + 25)
 
gp: K = bnfinit(x^10 - x^9 + 2*x^8 + 16*x^7 - 9*x^6 + 11*x^5 + 43*x^4 - 6*x^3 + 63*x^2 - 20*x + 25, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 2 x^{8} + 16 x^{7} - 9 x^{6} + 11 x^{5} + 43 x^{4} - 6 x^{3} + 63 x^{2} - 20 x + 25 \)

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:  \(-26439622160671=-\,31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.99$
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:  $31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(31\)
Dirichlet character group:    $\lbrace$$\chi_{31}(1,·)$, $\chi_{31}(2,·)$, $\chi_{31}(4,·)$, $\chi_{31}(8,·)$, $\chi_{31}(15,·)$, $\chi_{31}(16,·)$, $\chi_{31}(23,·)$, $\chi_{31}(27,·)$, $\chi_{31}(29,·)$, $\chi_{31}(30,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{8375} a^{9} + \frac{628}{8375} a^{8} - \frac{286}{8375} a^{7} - \frac{653}{8375} a^{6} - \frac{371}{8375} a^{5} - \frac{3873}{8375} a^{4} + \frac{2726}{8375} a^{3} + \frac{2798}{8375} a^{2} + \frac{251}{1675} a - \frac{49}{335}$

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

Class group and class number

$C_{3}$, which has order $3$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 485.913224212 \)
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

\(\Q(\sqrt{-31}) \), 5.5.923521.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$31$31.10.9.8$x^{10} + 521017$$10$$1$$9$$C_{10}$$[\ ]_{10}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.31.2t1.1c1$1$ $ 31 $ $x^{2} - x + 8$ $C_2$ (as 2T1) $1$ $-1$
* 1.31.5t1.1c1$1$ $ 31 $ $x^{5} - x^{4} - 12 x^{3} + 21 x^{2} + x - 5$ $C_5$ (as 5T1) $0$ $1$
* 1.31.10t1.1c1$1$ $ 31 $ $x^{10} - x^{9} + 2 x^{8} + 16 x^{7} - 9 x^{6} + 11 x^{5} + 43 x^{4} - 6 x^{3} + 63 x^{2} - 20 x + 25$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.31.5t1.1c2$1$ $ 31 $ $x^{5} - x^{4} - 12 x^{3} + 21 x^{2} + x - 5$ $C_5$ (as 5T1) $0$ $1$
* 1.31.10t1.1c2$1$ $ 31 $ $x^{10} - x^{9} + 2 x^{8} + 16 x^{7} - 9 x^{6} + 11 x^{5} + 43 x^{4} - 6 x^{3} + 63 x^{2} - 20 x + 25$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.31.5t1.1c3$1$ $ 31 $ $x^{5} - x^{4} - 12 x^{3} + 21 x^{2} + x - 5$ $C_5$ (as 5T1) $0$ $1$
* 1.31.10t1.1c3$1$ $ 31 $ $x^{10} - x^{9} + 2 x^{8} + 16 x^{7} - 9 x^{6} + 11 x^{5} + 43 x^{4} - 6 x^{3} + 63 x^{2} - 20 x + 25$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.31.5t1.1c4$1$ $ 31 $ $x^{5} - x^{4} - 12 x^{3} + 21 x^{2} + x - 5$ $C_5$ (as 5T1) $0$ $1$
* 1.31.10t1.1c4$1$ $ 31 $ $x^{10} - x^{9} + 2 x^{8} + 16 x^{7} - 9 x^{6} + 11 x^{5} + 43 x^{4} - 6 x^{3} + 63 x^{2} - 20 x + 25$ $C_{10}$ (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 *.