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Properties

Label 10.0.43684617614...9375.3
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{16}\cdot 31^{5}$
Root discriminant $73.12$
Ramified primes $5, 31$
Class number $1965$ (GRH)
Class group $[1965]$ (GRH)
Galois group $C_{10}$ (as 10T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![192493, -73895, 65760, -18165, 8050, -1494, 520, -80, 30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 + 30*x^8 - 80*x^7 + 520*x^6 - 1494*x^5 + 8050*x^4 - 18165*x^3 + 65760*x^2 - 73895*x + 192493)
 
gp: K = bnfinit(x^10 - 5*x^9 + 30*x^8 - 80*x^7 + 520*x^6 - 1494*x^5 + 8050*x^4 - 18165*x^3 + 65760*x^2 - 73895*x + 192493, 1)
 

Normalized defining polynomial

\( x^{10} - 5 x^{9} + 30 x^{8} - 80 x^{7} + 520 x^{6} - 1494 x^{5} + 8050 x^{4} - 18165 x^{3} + 65760 x^{2} - 73895 x + 192493 \)

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:  \(-4368461761474609375=-\,5^{16}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.12$
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 Galois and abelian over $\Q$.
Conductor:  \(775=5^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{775}(1,·)$, $\chi_{775}(681,·)$, $\chi_{775}(621,·)$, $\chi_{775}(526,·)$, $\chi_{775}(466,·)$, $\chi_{775}(371,·)$, $\chi_{775}(311,·)$, $\chi_{775}(216,·)$, $\chi_{775}(156,·)$, $\chi_{775}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{49} a^{8} + \frac{2}{49} a^{7} + \frac{3}{49} a^{6} + \frac{6}{49} a^{5} - \frac{2}{49} a^{4} + \frac{10}{49} a^{3} + \frac{19}{49} a^{2} - \frac{18}{49} a - \frac{3}{7}$, $\frac{1}{211902445704299} a^{9} + \frac{2116478523456}{211902445704299} a^{8} - \frac{14134110260132}{211902445704299} a^{7} + \frac{6972340283729}{211902445704299} a^{6} - \frac{3219402592479}{211902445704299} a^{5} + \frac{45893284703092}{211902445704299} a^{4} + \frac{74433258485657}{211902445704299} a^{3} - \frac{9547257663971}{211902445704299} a^{2} + \frac{32184323327530}{211902445704299} a + \frac{7640448223867}{30271777957757}$

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

Class group and class number

$C_{1965}$, which has order $1965$ (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:  \( -1 \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 257.113789169 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}$ (as 10T1):

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

Intermediate fields

\(\Q(\sqrt{-31}) \), 5.5.390625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ 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$5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
$31$31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$