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Properties

Label 10.0.75479754251...6687.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,11^{9}\cdot 317^{5}$
Root discriminant $154.09$
Ramified primes $11, 317$
Class number $180628$ (GRH)
Class group $[180628]$ (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![36028114321, -2180493932, 2180493932, -38239477, 38239477, -275474, 275474, -870, 870, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 870*x^8 - 870*x^7 + 275474*x^6 - 275474*x^5 + 38239477*x^4 - 38239477*x^3 + 2180493932*x^2 - 2180493932*x + 36028114321)
 
gp: K = bnfinit(x^10 - x^9 + 870*x^8 - 870*x^7 + 275474*x^6 - 275474*x^5 + 38239477*x^4 - 38239477*x^3 + 2180493932*x^2 - 2180493932*x + 36028114321, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 870 x^{8} - 870 x^{7} + 275474 x^{6} - 275474 x^{5} + 38239477 x^{4} - 38239477 x^{3} + 2180493932 x^{2} - 2180493932 x + 36028114321 \)

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:  \(-7547975425189709416687=-\,11^{9}\cdot 317^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $154.09$
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:  $11, 317$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3487=11\cdot 317\)
Dirichlet character group:    $\lbrace$$\chi_{3487}(1,·)$, $\chi_{3487}(3171,·)$, $\chi_{3487}(2854,·)$, $\chi_{3487}(2218,·)$, $\chi_{3487}(2220,·)$, $\chi_{3487}(1267,·)$, $\chi_{3487}(1269,·)$, $\chi_{3487}(633,·)$, $\chi_{3487}(316,·)$, $\chi_{3487}(3486,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3678739439} a^{6} - \frac{231660088}{3678739439} a^{5} + \frac{474}{3678739439} a^{4} + \frac{462751215}{3678739439} a^{3} + \frac{56169}{3678739439} a^{2} - \frac{230048405}{3678739439} a + \frac{986078}{3678739439}$, $\frac{1}{3678739439} a^{7} + \frac{553}{3678739439} a^{5} - \frac{92550243}{3678739439} a^{4} + \frac{87374}{3678739439} a^{3} + \frac{184038724}{3678739439} a^{2} + \frac{3451273}{3678739439} a - \frac{87949280}{3678739439}$, $\frac{1}{3678739439} a^{8} - \frac{740401944}{3678739439} a^{5} - \frac{174748}{3678739439} a^{4} + \frac{1794377559}{3678739439} a^{3} - \frac{27610184}{3678739439} a^{2} - \frac{1627061680}{3678739439} a - \frac{545301134}{3678739439}$, $\frac{1}{3678739439} a^{9} - \frac{224676}{3678739439} a^{5} - \frac{414087129}{3678739439} a^{4} - \frac{47331744}{3678739439} a^{3} + \frac{1539112400}{3678739439} a^{2} + \frac{1575435065}{3678739439} a + \frac{402853375}{3678739439}$

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

Class group and class number

$C_{180628}$, which has order $180628$ (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

\(\Q(\sqrt{-3487}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
317Data not computed