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Properties

Label 10.0.30094051633...1875.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 5^{5}\cdot 7^{5}\cdot 11^{9}$
Root discriminant $88.68$
Ramified primes $3, 5, 7, 11$
Class number $11528$ (GRH)
Class group $[2, 2, 2882]$ (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![157212199, -26517063, 26517063, -1383383, 1383383, -30031, 30031, -287, 287, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 287*x^8 - 287*x^7 + 30031*x^6 - 30031*x^5 + 1383383*x^4 - 1383383*x^3 + 26517063*x^2 - 26517063*x + 157212199)
 
gp: K = bnfinit(x^10 - x^9 + 287*x^8 - 287*x^7 + 30031*x^6 - 30031*x^5 + 1383383*x^4 - 1383383*x^3 + 26517063*x^2 - 26517063*x + 157212199, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 287 x^{8} - 287 x^{7} + 30031 x^{6} - 30031 x^{5} + 1383383 x^{4} - 1383383 x^{3} + 26517063 x^{2} - 26517063 x + 157212199 \)

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:  \(-30094051633627471875=-\,3^{5}\cdot 5^{5}\cdot 7^{5}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.68$
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:  $3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1155=3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1155}(1,·)$, $\chi_{1155}(1154,·)$, $\chi_{1155}(421,·)$, $\chi_{1155}(841,·)$, $\chi_{1155}(524,·)$, $\chi_{1155}(526,·)$, $\chi_{1155}(629,·)$, $\chi_{1155}(631,·)$, $\chi_{1155}(314,·)$, $\chi_{1155}(734,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{19370339} a^{6} - \frac{3668822}{19370339} a^{5} + \frac{156}{19370339} a^{4} + \frac{7311615}{19370339} a^{3} + \frac{6084}{19370339} a^{2} - \frac{3601400}{19370339} a + \frac{35152}{19370339}$, $\frac{1}{19370339} a^{7} + \frac{182}{19370339} a^{5} - \frac{1462323}{19370339} a^{4} + \frac{9464}{19370339} a^{3} + \frac{2881120}{19370339} a^{2} + \frac{123032}{19370339} a - \frac{1286118}{19370339}$, $\frac{1}{19370339} a^{8} + \frac{7671755}{19370339} a^{5} - \frac{18928}{19370339} a^{4} + \frac{8720581}{19370339} a^{3} - \frac{984256}{19370339} a^{2} - \frac{4422844}{19370339} a - \frac{6397664}{19370339}$, $\frac{1}{19370339} a^{9} - \frac{24336}{19370339} a^{5} - \frac{6482520}{19370339} a^{4} - \frac{1687296}{19370339} a^{3} + \frac{3136726}{19370339} a^{2} - \frac{5306365}{19370339} a - \frac{3672202}{19370339}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2882}$, which has order $11528$ (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

\(\Q(\sqrt{-1155}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ R R R R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$