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Properties

Label 10.0.31205881617...9375.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 5^{5}\cdot 11^{8}\cdot 61^{8}$
Root discriminant $707.01$
Ramified primes $3, 5, 11, 61$
Class number $13816550$ (GRH)
Class group $[5, 2763310]$ (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![95024896, 11268672, 24160144, 2918668, 1150624, -773019, 69265, 3602, -514, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 - 514*x^8 + 3602*x^7 + 69265*x^6 - 773019*x^5 + 1150624*x^4 + 2918668*x^3 + 24160144*x^2 + 11268672*x + 95024896)
 
gp: K = bnfinit(x^10 - 3*x^9 - 514*x^8 + 3602*x^7 + 69265*x^6 - 773019*x^5 + 1150624*x^4 + 2918668*x^3 + 24160144*x^2 + 11268672*x + 95024896, 1)
 

Normalized defining polynomial

\( x^{10} - 3 x^{9} - 514 x^{8} + 3602 x^{7} + 69265 x^{6} - 773019 x^{5} + 1150624 x^{4} + 2918668 x^{3} + 24160144 x^{2} + 11268672 x + 95024896 \)

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:  \(-31205881617963063819444759375=-\,3^{5}\cdot 5^{5}\cdot 11^{8}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $707.01$
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:  $3, 5, 11, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(10065=3\cdot 5\cdot 11\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{10065}(1,·)$, $\chi_{10065}(691,·)$, $\chi_{10065}(2986,·)$, $\chi_{10065}(3974,·)$, $\chi_{10065}(4426,·)$, $\chi_{10065}(5369,·)$, $\chi_{10065}(6059,·)$, $\chi_{10065}(8354,·)$, $\chi_{10065}(8671,·)$, $\chi_{10065}(9794,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{6} + \frac{1}{32} a^{5} - \frac{1}{32} a^{4} + \frac{3}{32} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{3424} a^{7} + \frac{51}{3424} a^{6} + \frac{193}{3424} a^{5} - \frac{79}{3424} a^{4} - \frac{185}{1712} a^{3} + \frac{77}{856} a^{2} - \frac{31}{428} a - \frac{26}{107}$, $\frac{1}{54784} a^{8} - \frac{1}{13696} a^{7} + \frac{299}{27392} a^{6} + \frac{483}{13696} a^{5} - \frac{2659}{54784} a^{4} - \frac{1029}{6848} a^{3} - \frac{3013}{13696} a^{2} - \frac{161}{856} a - \frac{355}{856}$, $\frac{1}{21286743131564615168} a^{9} - \frac{59087708012695}{21286743131564615168} a^{8} + \frac{370554593211521}{10643371565782307584} a^{7} - \frac{126511294329146603}{10643371565782307584} a^{6} + \frac{1063071695706087737}{21286743131564615168} a^{5} - \frac{512376318520273167}{21286743131564615168} a^{4} + \frac{186219892612161057}{5321685782891153792} a^{3} - \frac{449602412899828545}{5321685782891153792} a^{2} - \frac{9548221342964341}{83151340357674278} a + \frac{64396696626737441}{332605361430697112}$

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

Class group and class number

$C_{5}\times C_{2763310}$, which has order $13816550$ (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:  \( 400186.6439256542 \) (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

\(\Q(\sqrt{-15}) \), 5.5.202716958081.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{10}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$61$61.5.4.2$x^{5} + 122$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.2$x^{5} + 122$$5$$1$$4$$C_5$$[\ ]_{5}$