Properties

 Label 10.2.5611284433.1 Degree $10$ Signature $[2, 4]$ Discriminant $5611284433$ Root discriminant $$9.44$$ Ramified prime see page Class number $1$ Class group trivial Galois group $S_5$ (as 10T13)

Related objects

Show commands: SageMath / Pari/GP / Magma

Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + x^8 - 2*x^7 + x^6 + x^5 - 4*x^4 + 3*x^3 - 2*x + 1)

gp: K = bnfinit(x^10 - x^9 + x^8 - 2*x^7 + x^6 + x^5 - 4*x^4 + 3*x^3 - 2*x + 1, 1)

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

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

Invariants

 Degree: $10$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$5611284433$$ 5611284433 $$\medspace = 1777^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$9.44$$ 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: $$1777$$ 1777 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $1$ 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);

 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: $5$ 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: $a^{9}-a^{8}+a^{7}-2a^{6}+a^{5}+a^{4}-4a^{3}+3a^{2}-2$, $2a^{9}-2a^{8}+3a^{7}-6a^{6}+4a^{5}-2a^{4}-4a^{3}+4a^{2}-2a$, $a^{7}+a^{5}-2a^{4}-2a$, $2a^{9}-2a^{8}+3a^{7}-6a^{6}+4a^{5}-2a^{4}-4a^{3}+4a^{2}-3a$, $a^{9}-3a^{8}+3a^{7}-5a^{6}+7a^{5}-3a^{4}-2a^{3}+7a^{2}-4a$ a^9 - a^8 + a^7 - 2*a^6 + a^5 + a^4 - 4*a^3 + 3*a^2 - 2, 2*a^9 - 2*a^8 + 3*a^7 - 6*a^6 + 4*a^5 - 2*a^4 - 4*a^3 + 4*a^2 - 2*a, a^7 + a^5 - 2*a^4 - 2*a, 2*a^9 - 2*a^8 + 3*a^7 - 6*a^6 + 4*a^5 - 2*a^4 - 4*a^3 + 4*a^2 - 3*a, a^9 - 3*a^8 + 3*a^7 - 5*a^6 + 7*a^5 - 3*a^4 - 2*a^3 + 7*a^2 - 4*a sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$5.80208478511$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{4}\cdot 5.80208478511 \cdot 1}{2\sqrt{5611284433}}\approx 0.241436206187$

Galois group

$S_5$ (as 10T13):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 120 The 7 conjugacy class representatives for $S_5$ Character table for $S_5$

Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

 Degree 5 sibling: 5.1.1777.1 Degree 6 sibling: 6.2.5611284433.1 Degree 10 sibling: data not computed Degree 12 sibling: data not computed Degree 15 sibling: data not computed Degree 20 siblings: data not computed Degree 24 sibling: data not computed Degree 30 siblings: data not computed Degree 40 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 ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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
$$1777$$ $\Q_{1777}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1777}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2} Deg 2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$