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Properties

Label 10.0.13207350891...4208.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{10}\cdot 3^{5}\cdot 11^{8}\cdot 19^{5}$
Root discriminant $102.82$
Ramified primes $2, 3, 11, 19$
Class number $25420$ (GRH)
Class group $[2, 12710]$ (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![701996041, -23016264, 56562186, -1501924, 1872095, -37994, 31822, -442, 278, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 278*x^8 - 442*x^7 + 31822*x^6 - 37994*x^5 + 1872095*x^4 - 1501924*x^3 + 56562186*x^2 - 23016264*x + 701996041)
 
gp: K = bnfinit(x^10 - 2*x^9 + 278*x^8 - 442*x^7 + 31822*x^6 - 37994*x^5 + 1872095*x^4 - 1501924*x^3 + 56562186*x^2 - 23016264*x + 701996041, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} + 278 x^{8} - 442 x^{7} + 31822 x^{6} - 37994 x^{5} + 1872095 x^{4} - 1501924 x^{3} + 56562186 x^{2} - 23016264 x + 701996041 \)

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:  \(-132073508910190814208=-\,2^{10}\cdot 3^{5}\cdot 11^{8}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.82$
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:  $2, 3, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2508=2^{2}\cdot 3\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2508}(1,·)$, $\chi_{2508}(2051,·)$, $\chi_{2508}(229,·)$, $\chi_{2508}(455,·)$, $\chi_{2508}(2281,·)$, $\chi_{2508}(683,·)$, $\chi_{2508}(685,·)$, $\chi_{2508}(911,·)$, $\chi_{2508}(1367,·)$, $\chi_{2508}(1369,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{458913335509622053297117} a^{9} + \frac{27603524235643845890389}{458913335509622053297117} a^{8} + \frac{224317322755335449137053}{458913335509622053297117} a^{7} - \frac{55721365662476756676086}{458913335509622053297117} a^{6} - \frac{101932301851829382141927}{458913335509622053297117} a^{5} + \frac{50693247008010611550505}{458913335509622053297117} a^{4} - \frac{226903808371047171176192}{458913335509622053297117} a^{3} - \frac{215846983083652965498426}{458913335509622053297117} a^{2} - \frac{42705592265817023999037}{458913335509622053297117} a - \frac{2287868260839567147077}{19952753717809654491179}$

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

Class group and class number

$C_{2}\times C_{12710}$, which has order $25420$ (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

\(\Q(\sqrt{-57}) \), \(\Q(\zeta_{11})^+\)

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$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}$
$19$19.10.5.1$x^{10} - 722 x^{6} + 130321 x^{2} - 61902475$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$