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Properties

Label 10.0.28729302988...6875.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{16}\cdot 11^{9}\cdot 41^{8}$
Root discriminant $2217.34$
Ramified primes $5, 11, 41$
Class number $169497625$ (GRH)
Class group $[5, 5, 6779905]$ (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![85798288501893, 10961015191335, 1257274742625, 94284129720, 3706058675, 83279856, 696795, -6765, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 6765*x^7 + 696795*x^6 + 83279856*x^5 + 3706058675*x^4 + 94284129720*x^3 + 1257274742625*x^2 + 10961015191335*x + 85798288501893)
 
gp: K = bnfinit(x^10 - 6765*x^7 + 696795*x^6 + 83279856*x^5 + 3706058675*x^4 + 94284129720*x^3 + 1257274742625*x^2 + 10961015191335*x + 85798288501893, 1)
 

Normalized defining polynomial

\( x^{10} - 6765 x^{7} + 696795 x^{6} + 83279856 x^{5} + 3706058675 x^{4} + 94284129720 x^{3} + 1257274742625 x^{2} + 10961015191335 x + 85798288501893 \)

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:  \(-2872930298891221299684295654296875=-\,5^{16}\cdot 11^{9}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2217.34$
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:  $5, 11, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(11275=5^{2}\cdot 11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{11275}(1,·)$, $\chi_{11275}(1636,·)$, $\chi_{11275}(4321,·)$, $\chi_{11275}(4711,·)$, $\chi_{11275}(4856,·)$, $\chi_{11275}(6371,·)$, $\chi_{11275}(6816,·)$, $\chi_{11275}(10251,·)$, $\chi_{11275}(10916,·)$, $\chi_{11275}(11006,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{369} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{1107} a^{6} - \frac{1}{1107} a^{5} + \frac{4}{27} a^{4} + \frac{2}{27} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{3321} a^{7} + \frac{1}{3321} a^{6} - \frac{1}{1107} a^{5} - \frac{11}{81} a^{4} + \frac{13}{81} a^{3} + \frac{1}{9} a^{2} - \frac{1}{9} a$, $\frac{1}{222507} a^{8} - \frac{11}{74169} a^{7} + \frac{86}{222507} a^{6} + \frac{113}{222507} a^{5} - \frac{133}{1809} a^{4} - \frac{7}{5427} a^{3} - \frac{119}{603} a^{2} + \frac{262}{603} a + \frac{26}{67}$, $\frac{1}{49221865047646318294189988977168150461} a^{9} - \frac{53790105246847271323463268360514}{49221865047646318294189988977168150461} a^{8} + \frac{4874065851766940342300133714640690}{49221865047646318294189988977168150461} a^{7} - \frac{11904128407452739763417248617318979}{49221865047646318294189988977168150461} a^{6} + \frac{55337016175454571523952863319669515}{49221865047646318294189988977168150461} a^{5} - \frac{163851312766377310987235484044842634}{1200533293845032153516828999443125621} a^{4} + \frac{64276923421003850868555889702168052}{400177764615010717838942999814375207} a^{3} + \frac{23636725634713526010615942071953508}{133392588205003572612980999938125069} a^{2} - \frac{9137749313648082856831701644399543}{44464196068334524204326999979375023} a - \frac{192783243408696355712561986258174}{548940692201660792646012345424383}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{6779905}$, which has order $169497625$ (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:  \( 1487469.5436229876 \) (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

\(\Q(\sqrt{-11}) \), 5.5.16160924531640625.6

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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{10}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
$11$11.10.9.9$x^{10} + 297$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$41$41.10.8.3$x^{10} + 943 x^{5} + 242064$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$