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Properties

Label 10.0.207252522098163.3
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 31^{8}$
Root discriminant $27.02$
Ramified primes $3, 31$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_5$ (as 10T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![900, 1890, 1671, 888, 508, 164, 65, -22, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 14*x^8 - 22*x^7 + 65*x^6 + 164*x^5 + 508*x^4 + 888*x^3 + 1671*x^2 + 1890*x + 900)
 
gp: K = bnfinit(x^10 - 4*x^9 + 14*x^8 - 22*x^7 + 65*x^6 + 164*x^5 + 508*x^4 + 888*x^3 + 1671*x^2 + 1890*x + 900, 1)
 

Normalized defining polynomial

\( x^{10} - 4 x^{9} + 14 x^{8} - 22 x^{7} + 65 x^{6} + 164 x^{5} + 508 x^{4} + 888 x^{3} + 1671 x^{2} + 1890 x + 900 \)

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:  \(-207252522098163=-\,3^{5}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.02$
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:  $3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{60} a^{6} + \frac{7}{60} a^{4} + \frac{1}{60} a^{2} + \frac{1}{10} a$, $\frac{1}{180} a^{7} - \frac{1}{180} a^{6} - \frac{1}{60} a^{5} - \frac{17}{180} a^{4} + \frac{41}{180} a^{3} - \frac{1}{4} a^{2} + \frac{2}{15} a$, $\frac{1}{720} a^{8} + \frac{1}{720} a^{7} + \frac{1}{180} a^{6} + \frac{7}{720} a^{5} - \frac{1}{9} a^{4} - \frac{173}{720} a^{3} + \frac{1}{240} a^{2} - \frac{13}{120} a - \frac{1}{4}$, $\frac{1}{183600} a^{9} - \frac{11}{45900} a^{8} + \frac{499}{183600} a^{7} + \frac{1183}{183600} a^{6} - \frac{2107}{36720} a^{5} + \frac{8719}{183600} a^{4} + \frac{84}{425} a^{3} + \frac{8861}{61200} a^{2} + \frac{3877}{10200} a + \frac{389}{1020}$

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

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

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:  \( \frac{427}{183600} a^{9} - \frac{2213}{183600} a^{8} + \frac{2077}{45900} a^{7} - \frac{18119}{183600} a^{6} + \frac{227}{918} a^{5} + \frac{24493}{183600} a^{4} + \frac{17329}{20400} a^{3} + \frac{28411}{30600} a^{2} + \frac{9317}{5100} a + \frac{407}{255} \) (order $6$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8605.31566738 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5$ (as 10T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10
The 4 conjugacy class representatives for $D_5$
Character table for $D_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.8311689.1 x5

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

Sibling fields

Degree 5 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$31$31.5.4.4$x^{5} + 10633$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.4$x^{5} + 10633$$5$$1$$4$$C_5$$[\ ]_{5}$