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Properties

Label 10.0.16403967840...2863.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 11^{9}\cdot 31^{5}$
Root discriminant $83.46$
Ramified primes $3, 11, 31$
Class number $10576$ (GRH)
Class group $[2, 5288]$ (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![87151417, -16351644, 16351644, -960389, 960389, -23530, 23530, -254, 254, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 254*x^8 - 254*x^7 + 23530*x^6 - 23530*x^5 + 960389*x^4 - 960389*x^3 + 16351644*x^2 - 16351644*x + 87151417)
 
gp: K = bnfinit(x^10 - x^9 + 254*x^8 - 254*x^7 + 23530*x^6 - 23530*x^5 + 960389*x^4 - 960389*x^3 + 16351644*x^2 - 16351644*x + 87151417, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 254 x^{8} - 254 x^{7} + 23530 x^{6} - 23530 x^{5} + 960389 x^{4} - 960389 x^{3} + 16351644 x^{2} - 16351644 x + 87151417 \)

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:  \(-16403967840464902863=-\,3^{5}\cdot 11^{9}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.46$
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:  $3, 11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1023=3\cdot 11\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1023}(1,·)$, $\chi_{1023}(743,·)$, $\chi_{1023}(652,·)$, $\chi_{1023}(557,·)$, $\chi_{1023}(559,·)$, $\chi_{1023}(464,·)$, $\chi_{1023}(466,·)$, $\chi_{1023}(371,·)$, $\chi_{1023}(280,·)$, $\chi_{1023}(1022,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11074823} a^{6} - \frac{2366576}{11074823} a^{5} + \frac{138}{11074823} a^{4} + \frac{4714335}{11074823} a^{3} + \frac{4761}{11074823} a^{2} - \frac{2318525}{11074823} a + \frac{24334}{11074823}$, $\frac{1}{11074823} a^{7} + \frac{161}{11074823} a^{5} - \frac{942867}{11074823} a^{4} + \frac{7406}{11074823} a^{3} + \frac{1854820}{11074823} a^{2} + \frac{85169}{11074823} a - \frac{819216}{11074823}$, $\frac{1}{11074823} a^{8} + \frac{3531887}{11074823} a^{5} - \frac{14812}{11074823} a^{4} - \frac{4065151}{11074823} a^{3} - \frac{681352}{11074823} a^{2} - \frac{4080673}{11074823} a - \frac{3917774}{11074823}$, $\frac{1}{11074823} a^{9} - \frac{19044}{11074823} a^{5} - \frac{4173345}{11074823} a^{4} - \frac{1168032}{11074823} a^{3} + \frac{3261457}{11074823} a^{2} - \frac{4036591}{11074823} a - \frac{4311778}{11074823}$

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

Class group and class number

$C_{2}\times C_{5288}$, which has order $10576$ (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

\(\Q(\sqrt{-1023}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ R ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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}$$[\ ]_{2}^{5}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$31$31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$