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Properties

Label 10.0.11183262109375.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{8}\cdot 31^{5}$
Root discriminant $20.18$
Ramified primes $5, 31$
Class number $3$
Class group $[3]$
Galois group $D_5\times C_5$ (as 10T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -264, 430, -289, 27, 65, -24, -5, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 6*x^8 - 5*x^7 - 24*x^6 + 65*x^5 + 27*x^4 - 289*x^3 + 430*x^2 - 264*x + 64)
 
gp: K = bnfinit(x^10 - 3*x^9 + 6*x^8 - 5*x^7 - 24*x^6 + 65*x^5 + 27*x^4 - 289*x^3 + 430*x^2 - 264*x + 64, 1)
 

Normalized defining polynomial

\( x^{10} - 3 x^{9} + 6 x^{8} - 5 x^{7} - 24 x^{6} + 65 x^{5} + 27 x^{4} - 289 x^{3} + 430 x^{2} - 264 x + 64 \)

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:  \(-11183262109375=-\,5^{8}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.18$
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:  $5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{10} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{10} a^{4} - \frac{3}{10} a^{3} - \frac{2}{5} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{339800} a^{9} - \frac{11399}{339800} a^{8} - \frac{253}{33980} a^{7} - \frac{2045}{13592} a^{6} - \frac{34131}{84950} a^{5} + \frac{53269}{339800} a^{4} + \frac{67163}{339800} a^{3} + \frac{43643}{339800} a^{2} - \frac{28949}{169900} a + \frac{14703}{42475}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 1615.57595214 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{-31}) \)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ R ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$5.5.8.3$x^{5} - 5 x^{4} + 30$$5$$1$$8$$C_5$$[2]$
5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
$31$31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$