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Properties

Label 10.0.44336199890...9375.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 5^{5}\cdot 11^{9}\cdot 19^{5}$
Root discriminant $146.11$
Ramified primes $3, 5, 11, 19$
Class number $182200$ (GRH)
Class group $[2, 2, 45550]$ (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![21271947049, -1425424188, 1425424188, -27781733, 27781733, -222586, 222586, -782, 782, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 782*x^8 - 782*x^7 + 222586*x^6 - 222586*x^5 + 27781733*x^4 - 27781733*x^3 + 1425424188*x^2 - 1425424188*x + 21271947049)
 
gp: K = bnfinit(x^10 - x^9 + 782*x^8 - 782*x^7 + 222586*x^6 - 222586*x^5 + 27781733*x^4 - 27781733*x^3 + 1425424188*x^2 - 1425424188*x + 21271947049, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 782 x^{8} - 782 x^{7} + 222586 x^{6} - 222586 x^{5} + 27781733 x^{4} - 27781733 x^{3} + 1425424188 x^{2} - 1425424188 x + 21271947049 \)

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:  \(-4433619989050594959375=-\,3^{5}\cdot 5^{5}\cdot 11^{9}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.11$
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:  $3, 5, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3135=3\cdot 5\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3135}(1,·)$, $\chi_{3135}(2566,·)$, $\chi_{3135}(2279,·)$, $\chi_{3135}(2281,·)$, $\chi_{3135}(1996,·)$, $\chi_{3135}(1139,·)$, $\chi_{3135}(854,·)$, $\chi_{3135}(856,·)$, $\chi_{3135}(569,·)$, $\chi_{3135}(3134,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2198073239} a^{6} - \frac{153927872}{2198073239} a^{5} + \frac{426}{2198073239} a^{4} + \frac{307436415}{2198073239} a^{3} + \frac{45369}{2198073239} a^{2} - \frac{152746925}{2198073239} a + \frac{715822}{2198073239}$, $\frac{1}{2198073239} a^{7} + \frac{497}{2198073239} a^{5} - \frac{61487283}{2198073239} a^{4} + \frac{70574}{2198073239} a^{3} + \frac{122197540}{2198073239} a^{2} + \frac{2505377}{2198073239} a - \frac{58133808}{2198073239}$, $\frac{1}{2198073239} a^{8} - \frac{491898264}{2198073239} a^{5} - \frac{141148}{2198073239} a^{4} - \frac{1006647224}{2198073239} a^{3} - \frac{20043016}{2198073239} a^{2} - \frac{1075475448}{2198073239} a - \frac{355763534}{2198073239}$, $\frac{1}{2198073239} a^{9} - \frac{181476}{2198073239} a^{5} - \frac{274944465}{2198073239} a^{4} - \frac{34359456}{2198073239} a^{3} + \frac{1017341640}{2198073239} a^{2} + \frac{825842465}{2198073239} a + \frac{48904359}{2198073239}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{45550}$, which has order $182200$ (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

\(\Q(\sqrt{-3135}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.10.5.2$x^{10} - 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$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$19$19.10.5.1$x^{10} - 722 x^{6} + 130321 x^{2} - 61902475$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$