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Properties

Label 10.0.24123421133...9375.3
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{5}\cdot 11^{9}\cdot 41^{9}$
Root discriminant $547.32$
Ramified primes $5, 11, 41$
Class number $5071000$ (GRH)
Class group $[10, 507100]$ (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![1689938944, -1045275584, 157170784, -23104764, 8870412, -68025, 109561, 54, 474, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 474*x^8 + 54*x^7 + 109561*x^6 - 68025*x^5 + 8870412*x^4 - 23104764*x^3 + 157170784*x^2 - 1045275584*x + 1689938944)
 
gp: K = bnfinit(x^10 - x^9 + 474*x^8 + 54*x^7 + 109561*x^6 - 68025*x^5 + 8870412*x^4 - 23104764*x^3 + 157170784*x^2 - 1045275584*x + 1689938944, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 474 x^{8} + 54 x^{7} + 109561 x^{6} - 68025 x^{5} + 8870412 x^{4} - 23104764 x^{3} + 157170784 x^{2} - 1045275584 x + 1689938944 \)

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:  \(-2412342113372980700918909375=-\,5^{5}\cdot 11^{9}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $547.32$
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:  \(2255=5\cdot 11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{2255}(256,·)$, $\chi_{2255}(1,·)$, $\chi_{2255}(2114,·)$, $\chi_{2255}(141,·)$, $\chi_{2255}(2254,·)$, $\chi_{2255}(1999,·)$, $\chi_{2255}(16,·)$, $\chi_{2255}(1841,·)$, $\chi_{2255}(414,·)$, $\chi_{2255}(2239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{7} - \frac{1}{64} a^{6} + \frac{3}{64} a^{5} - \frac{3}{64} a^{4} - \frac{3}{16} a^{3} + \frac{3}{16} a^{2}$, $\frac{1}{256} a^{8} - \frac{1}{128} a^{7} - \frac{1}{32} a^{6} + \frac{3}{128} a^{5} - \frac{13}{256} a^{4} + \frac{7}{64} a^{3} - \frac{7}{64} a^{2} + \frac{1}{16} a$, $\frac{1}{28906056918767185274987320832} a^{9} - \frac{5624705592124123150635993}{3613257114845898159373415104} a^{8} + \frac{54622341342918607207460407}{7226514229691796318746830208} a^{7} + \frac{188156584758275075434804447}{14453028459383592637493660416} a^{6} + \frac{418120757518111870190192087}{28906056918767185274987320832} a^{5} - \frac{732218360582309057812659135}{14453028459383592637493660416} a^{4} - \frac{237263813944157734231329177}{7226514229691796318746830208} a^{3} + \frac{823513191555051025309336227}{3613257114845898159373415104} a^{2} + \frac{341248178013784352835845233}{903314278711474539843353776} a + \frac{17058304822712752023175612}{56457142419467158740209611}$

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

Class group and class number

$C_{10}\times C_{507100}$, which has order $5071000$ (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:  \( 371368.7529642051 \) (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

\(\Q(\sqrt{-2255}) \), 5.5.41371966801.4

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.3$x^{10} - 891$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$41$41.10.9.9$x^{10} + 11477376$$10$$1$$9$$C_{10}$$[\ ]_{10}$