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Properties

Label 10.0.12150000000...000.45
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{15}\cdot 3^{5}\cdot 5^{16}$
Root discriminant $64.34$
Ramified primes $2, 3, 5$
Class number $1010$ (GRH)
Class group $[1010]$ (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![68257, 8740, 22730, 3040, 3185, 158, 240, -10, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 10*x^8 - 10*x^7 + 240*x^6 + 158*x^5 + 3185*x^4 + 3040*x^3 + 22730*x^2 + 8740*x + 68257)
 
gp: K = bnfinit(x^10 + 10*x^8 - 10*x^7 + 240*x^6 + 158*x^5 + 3185*x^4 + 3040*x^3 + 22730*x^2 + 8740*x + 68257, 1)
 

Normalized defining polynomial

\( x^{10} + 10 x^{8} - 10 x^{7} + 240 x^{6} + 158 x^{5} + 3185 x^{4} + 3040 x^{3} + 22730 x^{2} + 8740 x + 68257 \)

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:  \(-1215000000000000000=-\,2^{15}\cdot 3^{5}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.34$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(600=2^{3}\cdot 3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{600}(1,·)$, $\chi_{600}(581,·)$, $\chi_{600}(481,·)$, $\chi_{600}(361,·)$, $\chi_{600}(461,·)$, $\chi_{600}(241,·)$, $\chi_{600}(341,·)$, $\chi_{600}(121,·)$, $\chi_{600}(221,·)$, $\chi_{600}(101,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{49} a^{8} + \frac{1}{49} a^{7} + \frac{3}{49} a^{6} - \frac{15}{49} a^{5} + \frac{5}{49} a^{4} - \frac{11}{49} a^{3} - \frac{2}{49} a^{2} - \frac{10}{49} a$, $\frac{1}{29185959915049} a^{9} - \frac{201322355236}{29185959915049} a^{8} - \frac{10702865471}{595631835001} a^{7} + \frac{1990656459677}{29185959915049} a^{6} - \frac{2233253187367}{29185959915049} a^{5} - \frac{1127009795574}{29185959915049} a^{4} + \frac{10294153852762}{29185959915049} a^{3} - \frac{6075357562180}{29185959915049} a^{2} - \frac{13752437758064}{29185959915049} a - \frac{68426177}{427589257}$

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

Class group and class number

$C_{1010}$, which has order $1010$ (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:  \( 257.113789169 \) (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

\(\Q(\sqrt{-6}) \), 5.5.390625.1

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 R R R ${\href{/LocalNumberField/7.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$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}$
$5$5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$