↑

Properties

Label 10.0.120560905159375.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{5}\cdot 131^{5}$
Root discriminant $25.59$
Ramified primes $5, 131$
Class number $12$
Class group $[12]$
Galois group $D_{10}$ (as 10T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1424, -276, -340, -55, 54, 67, -1, -13, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 7*x^8 - 13*x^7 - x^6 + 67*x^5 + 54*x^4 - 55*x^3 - 340*x^2 - 276*x + 1424)
 
gp: K = bnfinit(x^10 - 2*x^9 + 7*x^8 - 13*x^7 - x^6 + 67*x^5 + 54*x^4 - 55*x^3 - 340*x^2 - 276*x + 1424, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} + 7 x^{8} - 13 x^{7} - x^{6} + 67 x^{5} + 54 x^{4} - 55 x^{3} - 340 x^{2} - 276 x + 1424 \)

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:  \(-120560905159375=-\,5^{5}\cdot 131^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.59$
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:  $5, 131$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{356} a^{8} + \frac{43}{356} a^{7} - \frac{73}{356} a^{6} - \frac{53}{356} a^{5} - \frac{5}{356} a^{4} + \frac{15}{356} a^{3} - \frac{27}{178} a^{2} - \frac{34}{89} a$, $\frac{1}{663292792} a^{9} - \frac{76759}{82911599} a^{8} - \frac{36022845}{663292792} a^{7} + \frac{121357793}{663292792} a^{6} + \frac{158893825}{663292792} a^{5} + \frac{81912041}{663292792} a^{4} - \frac{7300360}{82911599} a^{3} - \frac{17110911}{663292792} a^{2} + \frac{49178771}{331646396} a + \frac{174218}{931591}$

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

Class group and class number

$C_{12}$, which has order $12$

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:  \( 458.38640509 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 10T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-655}) \), 5.1.17161.1

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

Sibling fields

Galois closure: data not computed
Degree 10 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$131$131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$