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Properties

Label 10.0.62846014137...4375.3
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{5}\cdot 11^{9}\cdot 31^{8}$
Root discriminant $301.88$
Ramified primes $5, 11, 31$
Class number $1615100$ (GRH)
Class group $[5, 323020]$ (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![259861504, 41505856, 13054024, -277572, -111526, 39445, 17637, 50, -200, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 - 200*x^8 + 50*x^7 + 17637*x^6 + 39445*x^5 - 111526*x^4 - 277572*x^3 + 13054024*x^2 + 41505856*x + 259861504)
 
gp: K = bnfinit(x^10 - 3*x^9 - 200*x^8 + 50*x^7 + 17637*x^6 + 39445*x^5 - 111526*x^4 - 277572*x^3 + 13054024*x^2 + 41505856*x + 259861504, 1)
 

Normalized defining polynomial

\( x^{10} - 3 x^{9} - 200 x^{8} + 50 x^{7} + 17637 x^{6} + 39445 x^{5} - 111526 x^{4} - 277572 x^{3} + 13054024 x^{2} + 41505856 x + 259861504 \)

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:  \(-6284601413776876558534375=-\,5^{5}\cdot 11^{9}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $301.88$
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:  $5, 11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1705=5\cdot 11\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1705}(1,·)$, $\chi_{1705}(1349,·)$, $\chi_{1705}(39,·)$, $\chi_{1705}(714,·)$, $\chi_{1705}(1614,·)$, $\chi_{1705}(1521,·)$, $\chi_{1705}(1461,·)$, $\chi_{1705}(566,·)$, $\chi_{1705}(1399,·)$, $\chi_{1705}(1566,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{1232} a^{5} + \frac{1}{308} a^{4} + \frac{5}{176} a^{3} - \frac{37}{308} a^{2} - \frac{127}{308} a - \frac{5}{11}$, $\frac{1}{2464} a^{6} - \frac{1}{2464} a^{5} + \frac{15}{2464} a^{4} + \frac{293}{2464} a^{3} + \frac{29}{308} a^{2} + \frac{3}{56} a - \frac{4}{11}$, $\frac{1}{9856} a^{7} - \frac{1}{9856} a^{6} - \frac{1}{9856} a^{5} - \frac{387}{9856} a^{4} - \frac{195}{1232} a^{3} - \frac{453}{2464} a^{2} + \frac{221}{616} a - \frac{1}{11}$, $\frac{1}{39424} a^{8} - \frac{3}{19712} a^{6} - \frac{1}{2464} a^{5} - \frac{1767}{39424} a^{4} + \frac{59}{2464} a^{3} + \frac{1675}{9856} a^{2} + \frac{615}{1232} a + \frac{1}{11}$, $\frac{1}{20660000023151104} a^{9} - \frac{227483354093}{20660000023151104} a^{8} - \frac{338921888847}{10330000011575552} a^{7} - \frac{1931240779621}{10330000011575552} a^{6} + \frac{6456921055297}{20660000023151104} a^{5} - \frac{108567965229389}{20660000023151104} a^{4} - \frac{994354442548305}{5165000005787776} a^{3} + \frac{841579324771233}{5165000005787776} a^{2} + \frac{20376549499991}{92232142960496} a - \frac{208066306871}{823501276433}$

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

Class group and class number

$C_{5}\times C_{323020}$, which has order $1615100$ (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:  \( 254846.18150922793 \) (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

\(\Q(\sqrt{-55}) \), 5.5.13521270961.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{10}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ R ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$31$31.5.4.3$x^{5} - 1519$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.3$x^{5} - 1519$$5$$1$$4$$C_5$$[\ ]_{5}$