↑

Properties

Label 10.0.36879041080...5159.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 11^{9}\cdot 23^{5}$
Root discriminant $71.89$
Ramified primes $3, 11, 23$
Class number $4224$ (GRH)
Class group $[2, 2, 2, 2, 2, 132]$ (GRH)
Galois group $C_{10}$ (as 10T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20603287, -4984860, 4984860, -391205, 391205, -12904, 12904, -188, 188, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 188*x^8 - 188*x^7 + 12904*x^6 - 12904*x^5 + 391205*x^4 - 391205*x^3 + 4984860*x^2 - 4984860*x + 20603287)
 
gp: K = bnfinit(x^10 - x^9 + 188*x^8 - 188*x^7 + 12904*x^6 - 12904*x^5 + 391205*x^4 - 391205*x^3 + 4984860*x^2 - 4984860*x + 20603287, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 188 x^{8} - 188 x^{7} + 12904 x^{6} - 12904 x^{5} + 391205 x^{4} - 391205 x^{3} + 4984860 x^{2} - 4984860 x + 20603287 \)

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:  \(-3687904108026165159=-\,3^{5}\cdot 11^{9}\cdot 23^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.89$
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:  $3, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(759=3\cdot 11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{759}(1,·)$, $\chi_{759}(68,·)$, $\chi_{759}(70,·)$, $\chi_{759}(553,·)$, $\chi_{759}(206,·)$, $\chi_{759}(689,·)$, $\chi_{759}(691,·)$, $\chi_{759}(758,·)$, $\chi_{759}(346,·)$, $\chi_{759}(413,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2852873} a^{6} - \frac{819890}{2852873} a^{5} + \frac{102}{2852873} a^{4} - \frac{1221698}{2852873} a^{3} + \frac{2601}{2852873} a^{2} - \frac{798755}{2852873} a + \frac{9826}{2852873}$, $\frac{1}{2852873} a^{7} + \frac{119}{2852873} a^{5} - \frac{326235}{2852873} a^{4} + \frac{4046}{2852873} a^{3} + \frac{639004}{2852873} a^{2} + \frac{34391}{2852873} a - \frac{274212}{2852873}$, $\frac{1}{2852873} a^{8} + \frac{242993}{2852873} a^{5} - \frac{8092}{2852873} a^{4} + \frac{524543}{2852873} a^{3} - \frac{275128}{2852873} a^{2} + \frac{632824}{2852873} a - \frac{1169294}{2852873}$, $\frac{1}{2852873} a^{9} - \frac{10404}{2852873} a^{5} + \frac{1415114}{2852873} a^{4} - \frac{471648}{2852873} a^{3} - \frac{907036}{2852873} a^{2} + \frac{1195612}{2852873} a + \frac{205483}{2852873}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{132}$, which has order $4224$ (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:  \( 26.1711060094 \) (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

\(\Q(\sqrt{-759}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$