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Properties

Label 10.0.93952943605...4375.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{5}\cdot 11^{8}\cdot 107^{5}$
Root discriminant $157.50$
Ramified primes $5, 11, 107$
Class number $185570$ (GRH)
Class group $[185570]$ (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![45829332361, -981671945, 1655453824, -28731717, 24167966, -318692, 178230, -1588, 664, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 664*x^8 - 1588*x^7 + 178230*x^6 - 318692*x^5 + 24167966*x^4 - 28731717*x^3 + 1655453824*x^2 - 981671945*x + 45829332361)
 
gp: K = bnfinit(x^10 - 3*x^9 + 664*x^8 - 1588*x^7 + 178230*x^6 - 318692*x^5 + 24167966*x^4 - 28731717*x^3 + 1655453824*x^2 - 981671945*x + 45829332361, 1)
 

Normalized defining polynomial

\( x^{10} - 3 x^{9} + 664 x^{8} - 1588 x^{7} + 178230 x^{6} - 318692 x^{5} + 24167966 x^{4} - 28731717 x^{3} + 1655453824 x^{2} - 981671945 x + 45829332361 \)

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:  \(-9395294360545792084375=-\,5^{5}\cdot 11^{8}\cdot 107^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $157.50$
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:  $5, 11, 107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(5885=5\cdot 11\cdot 107\)
Dirichlet character group:    $\lbrace$$\chi_{5885}(3744,·)$, $\chi_{5885}(1,·)$, $\chi_{5885}(1604,·)$, $\chi_{5885}(5349,·)$, $\chi_{5885}(5351,·)$, $\chi_{5885}(1071,·)$, $\chi_{5885}(4816,·)$, $\chi_{5885}(2674,·)$, $\chi_{5885}(2676,·)$, $\chi_{5885}(2139,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2084451122825173561287307711} a^{9} + \frac{267958467655649385307666562}{2084451122825173561287307711} a^{8} - \frac{266328298741651207339998515}{2084451122825173561287307711} a^{7} + \frac{874247428113994758770770471}{2084451122825173561287307711} a^{6} + \frac{664473097688303282341599117}{2084451122825173561287307711} a^{5} - \frac{931764774242993069976354202}{2084451122825173561287307711} a^{4} - \frac{513897052899702436426046126}{2084451122825173561287307711} a^{3} + \frac{16063108183293947567353832}{48475607507562175843890877} a^{2} - \frac{640445639543460967286517154}{2084451122825173561287307711} a - \frac{10421893303853243794112247}{48475607507562175843890877}$

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

Class group and class number

$C_{185570}$, which has order $185570$ (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

\(\Q(\sqrt{-535}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$107$107.10.5.1$x^{10} - 22898 x^{6} + 131079601 x^{2} - 1136066901867$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$