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Properties

Label 10.0.28687182077...5408.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{15}\cdot 3^{5}\cdot 7^{5}\cdot 11^{8}$
Root discriminant $88.26$
Ramified primes $2, 3, 7, 11$
Class number $13684$ (GRH)
Class group $[2, 6842]$ (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![160833331, -6992754, 17082831, -603934, 751955, -20444, 17152, -322, 203, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 203*x^8 - 322*x^7 + 17152*x^6 - 20444*x^5 + 751955*x^4 - 603934*x^3 + 17082831*x^2 - 6992754*x + 160833331)
 
gp: K = bnfinit(x^10 - 2*x^9 + 203*x^8 - 322*x^7 + 17152*x^6 - 20444*x^5 + 751955*x^4 - 603934*x^3 + 17082831*x^2 - 6992754*x + 160833331, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} + 203 x^{8} - 322 x^{7} + 17152 x^{6} - 20444 x^{5} + 751955 x^{4} - 603934 x^{3} + 17082831 x^{2} - 6992754 x + 160833331 \)

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:  \(-28687182077984145408=-\,2^{15}\cdot 3^{5}\cdot 7^{5}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.26$
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:  $2, 3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1848=2^{3}\cdot 3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1848}(1,·)$, $\chi_{1848}(419,·)$, $\chi_{1848}(1345,·)$, $\chi_{1848}(169,·)$, $\chi_{1848}(587,·)$, $\chi_{1848}(841,·)$, $\chi_{1848}(1681,·)$, $\chi_{1848}(1763,·)$, $\chi_{1848}(1259,·)$, $\chi_{1848}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{23} a^{8} + \frac{1}{23} a^{7} - \frac{5}{23} a^{6} + \frac{4}{23} a^{5} + \frac{3}{23} a^{4} - \frac{4}{23} a^{3} - \frac{8}{23} a^{2} - \frac{8}{23} a + \frac{3}{23}$, $\frac{1}{1005779362296704479199} a^{9} - \frac{20587894331169669249}{1005779362296704479199} a^{8} - \frac{21422631406060620181}{1005779362296704479199} a^{7} - \frac{171848424374652896407}{1005779362296704479199} a^{6} + \frac{60023367276450832506}{1005779362296704479199} a^{5} + \frac{128112757953260956606}{1005779362296704479199} a^{4} + \frac{256268377316273195225}{1005779362296704479199} a^{3} - \frac{427922982746170869053}{1005779362296704479199} a^{2} + \frac{238697419866040739818}{1005779362296704479199} a + \frac{326628375625331581939}{1005779362296704479199}$

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

Class group and class number

$C_{2}\times C_{6842}$, which has order $13684$ (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

\(\Q(\sqrt{-42}) \), \(\Q(\zeta_{11})^+\)

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }$ R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$