Properties

Label 10.0.400000000.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{10}\cdot 5^{8}$
Root discriminant $7.25$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $D_5\times C_5$ (as 10T6)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 9*x^8 - 14*x^7 + 15*x^6 - 10*x^5 + 3*x^4 + 2*x^3 - 2*x^2 + 1)
 
gp: K = bnfinit(x^10 - 4*x^9 + 9*x^8 - 14*x^7 + 15*x^6 - 10*x^5 + 3*x^4 + 2*x^3 - 2*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -2, 2, 3, -10, 15, -14, 9, -4, 1]);
 

Normalized defining polynomial

\( x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1 \)

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

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-400000000=-\,2^{10}\cdot 5^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $7.25$
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:  $2, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $5$
This field is not Galois over $\Q$.
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);
 

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:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( a^{8} - 3 a^{7} + 5 a^{6} - 6 a^{5} + 5 a^{4} - 2 a^{3} + a^{2} \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{9} - 3 a^{8} + 5 a^{7} - 6 a^{6} + 5 a^{5} - 2 a^{4} + a^{3} \),  \( a^{8} - 3 a^{7} + 6 a^{6} - 8 a^{5} + 7 a^{4} - 4 a^{3} + a^{2} + a \),  \( a^{9} - 3 a^{8} + 6 a^{7} - 8 a^{6} + 6 a^{5} - 2 a^{4} - a^{3} + 2 a^{2} + a - 1 \),  \( a^{6} - 2 a^{5} + 2 a^{4} - a^{3} - a^{2} + 2 a \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1.83575423504 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_5\times D_5$ (as 10T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 20 conjugacy class representatives for $D_5\times C_5$
Character table for $D_5\times C_5$

Intermediate fields

\(\Q(\sqrt{-1}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 sibling: data not computed
Degree 25 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$5$5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.25.5t1.a.c$1$ $ 5^{2}$ $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.25.5t1.a.b$1$ $ 5^{2}$ $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.100.10t1.a.c$1$ $ 2^{2} \cdot 5^{2}$ $x^{10} + 20 x^{8} + 120 x^{6} + 225 x^{4} + 90 x^{2} + 1$ $C_{10}$ (as 10T1) $0$ $-1$
1.25.5t1.a.a$1$ $ 5^{2}$ $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.100.10t1.a.d$1$ $ 2^{2} \cdot 5^{2}$ $x^{10} + 20 x^{8} + 120 x^{6} + 225 x^{4} + 90 x^{2} + 1$ $C_{10}$ (as 10T1) $0$ $-1$
1.100.10t1.a.b$1$ $ 2^{2} \cdot 5^{2}$ $x^{10} + 20 x^{8} + 120 x^{6} + 225 x^{4} + 90 x^{2} + 1$ $C_{10}$ (as 10T1) $0$ $-1$
1.25.5t1.a.d$1$ $ 5^{2}$ $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.100.10t1.a.a$1$ $ 2^{2} \cdot 5^{2}$ $x^{10} + 20 x^{8} + 120 x^{6} + 225 x^{4} + 90 x^{2} + 1$ $C_{10}$ (as 10T1) $0$ $-1$
2.2500.10t6.b.d$2$ $ 2^{2} \cdot 5^{4}$ $x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$ $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.100.10t6.b.c$2$ $ 2^{2} \cdot 5^{2}$ $x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$ $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.100.10t6.b.d$2$ $ 2^{2} \cdot 5^{2}$ $x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$ $D_5\times C_5$ (as 10T6) $0$ $0$
2.2500.5t2.a.a$2$ $ 2^{2} \cdot 5^{4}$ $x^{5} + 10 x^{3} - 10 x^{2} - 15 x - 18$ $D_{5}$ (as 5T2) $1$ $0$
2.2500.5t2.a.b$2$ $ 2^{2} \cdot 5^{4}$ $x^{5} + 10 x^{3} - 10 x^{2} - 15 x - 18$ $D_{5}$ (as 5T2) $1$ $0$
* 2.100.10t6.b.a$2$ $ 2^{2} \cdot 5^{2}$ $x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$ $D_5\times C_5$ (as 10T6) $0$ $0$
2.2500.10t6.b.c$2$ $ 2^{2} \cdot 5^{4}$ $x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$ $D_5\times C_5$ (as 10T6) $0$ $0$
2.2500.10t6.b.a$2$ $ 2^{2} \cdot 5^{4}$ $x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$ $D_5\times C_5$ (as 10T6) $0$ $0$
2.2500.10t6.b.b$2$ $ 2^{2} \cdot 5^{4}$ $x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$ $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.100.10t6.b.b$2$ $ 2^{2} \cdot 5^{2}$ $x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$ $D_5\times C_5$ (as 10T6) $0$ $0$

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 *.