# Properties

 Label 10.0.224415603.1 Degree $10$ Signature $[0, 5]$ Discriminant $-224415603$ Root discriminant $6.84$ Ramified primes $3, 31$ Class number $1$ Class group trivial Galois group $D_5\times C_5$ (as 10T6)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 2*x^8 - 3*x^7 + 3*x^6 - 7*x^5 + 8*x^4 - 7*x^3 + 7*x^2 - 4*x + 1)

gp: K = bnfinit(x^10 + 2*x^8 - 3*x^7 + 3*x^6 - 7*x^5 + 8*x^4 - 7*x^3 + 7*x^2 - 4*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 7, -7, 8, -7, 3, -3, 2, 0, 1]);

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

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: $$-224415603$$$$\medspace = -\,3^{5}\cdot 31^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $6.84$ 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: $3, 31$ 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: $$2 a^{9} + a^{8} + 3 a^{7} - 4 a^{6} + 3 a^{5} - 8 a^{4} + 8 a^{3} - 4 a^{2} + 4 a - 1$$ (order $6$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{9} + 3 a^{7} - 2 a^{6} + 4 a^{5} - 9 a^{4} + 7 a^{3} - 9 a^{2} + 8 a - 2$$,  $$2 a^{9} + 2 a^{8} + 4 a^{7} - 3 a^{6} + a^{5} - 9 a^{4} + 5 a^{3} - 3 a^{2} + 5 a - 2$$,  $$a^{7} + a^{5} - 2 a^{4} + 3 a^{3} - 3 a^{2} + 4 a - 2$$,  $$2 a^{9} + 2 a^{8} + 5 a^{7} - 3 a^{6} + 2 a^{5} - 11 a^{4} + 8 a^{3} - 6 a^{2} + 10 a - 4$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$1.78149530531$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{5}\cdot 1.78149530531 \cdot 1}{6\sqrt{224415603}}\approx 0.194091380560$

## 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

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 ${\href{/padicField/2.10.0.1}{10} }$ R ${\href{/padicField/5.2.0.1}{2} }^{5}$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }$ ${\href{/padicField/29.10.0.1}{10} }$ R ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ ${\href{/padicField/47.10.0.1}{10} }$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.10.0.1}{10} }$

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}$$[\ ]_{2}^{5} 3131.5.4.5x^{5} - 74431$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $3$ $$\Q(\sqrt{-3})$$ $C_2$ (as 2T1) $1$ $-1$
1.93.10t1.a.d$1$ $3 \cdot 31$ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.31.5t1.a.d$1$ $31$ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
1.93.10t1.a.b$1$ $3 \cdot 31$ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.31.5t1.a.b$1$ $31$ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
1.93.10t1.a.c$1$ $3 \cdot 31$ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.93.10t1.a.a$1$ $3 \cdot 31$ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.31.5t1.a.a$1$ $31$ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
1.31.5t1.a.c$1$ $31$ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
2.2883.5t2.a.b$2$ $3 \cdot 31^{2}$ 5.1.8311689.1 $D_{5}$ (as 5T2) $1$ $0$
2.2883.10t6.b.a$2$ $3 \cdot 31^{2}$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.2883.10t6.b.b$2$ $3 \cdot 31^{2}$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.93.10t6.b.b$2$ $3 \cdot 31$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.93.10t6.b.a$2$ $3 \cdot 31$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.2883.10t6.b.c$2$ $3 \cdot 31^{2}$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.93.10t6.b.c$2$ $3 \cdot 31$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
2.2883.5t2.a.a$2$ $3 \cdot 31^{2}$ 5.1.8311689.1 $D_{5}$ (as 5T2) $1$ $0$
2.2883.10t6.b.d$2$ $3 \cdot 31^{2}$ 10.0.224415603.1 $D_5\times C_5$ (as 10T6) $0$ $0$
* 2.93.10t6.b.d$2$ $3 \cdot 31$ 10.0.224415603.1 $D_5\times C_5$ (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 *.