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Properties

Label 10.0.40494828100...9375.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{17}\cdot 11^{8}\cdot 19^{5}$
Root discriminant $457.87$
Ramified primes $5, 11, 19$
Class number $7216000$ (GRH)
Class group $[2, 2, 10, 10, 18040]$ (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![533780224, 38261440, 21008400, 1396580, 251560, 29011, 11105, -50, -90, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 - 90*x^8 - 50*x^7 + 11105*x^6 + 29011*x^5 + 251560*x^4 + 1396580*x^3 + 21008400*x^2 + 38261440*x + 533780224)
 
gp: K = bnfinit(x^10 - 5*x^9 - 90*x^8 - 50*x^7 + 11105*x^6 + 29011*x^5 + 251560*x^4 + 1396580*x^3 + 21008400*x^2 + 38261440*x + 533780224, 1)
 

Normalized defining polynomial

\( x^{10} - 5 x^{9} - 90 x^{8} - 50 x^{7} + 11105 x^{6} + 29011 x^{5} + 251560 x^{4} + 1396580 x^{3} + 21008400 x^{2} + 38261440 x + 533780224 \)

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:  \(-404948281009841156005859375=-\,5^{17}\cdot 11^{8}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $457.87$
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:  $5, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(5225=5^{2}\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{5225}(1,·)$, $\chi_{5225}(379,·)$, $\chi_{5225}(474,·)$, $\chi_{5225}(664,·)$, $\chi_{5225}(856,·)$, $\chi_{5225}(1236,·)$, $\chi_{5225}(1996,·)$, $\chi_{5225}(2566,·)$, $\chi_{5225}(3419,·)$, $\chi_{5225}(4084,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{200} a^{5} + \frac{1}{20} a^{4} - \frac{9}{40} a^{3} - \frac{1}{4} a^{2} - \frac{3}{10} a - \frac{4}{25}$, $\frac{1}{800} a^{6} - \frac{1}{800} a^{5} - \frac{1}{160} a^{4} - \frac{31}{160} a^{3} + \frac{7}{40} a^{2} + \frac{57}{200} a - \frac{3}{50}$, $\frac{1}{800} a^{7} - \frac{1}{400} a^{5} - \frac{1}{40} a^{4} + \frac{1}{160} a^{3} + \frac{17}{200} a^{2} - \frac{13}{40} a - \frac{11}{50}$, $\frac{1}{121600} a^{8} - \frac{17}{30400} a^{7} + \frac{31}{60800} a^{6} - \frac{3}{1600} a^{5} - \frac{1151}{24320} a^{4} + \frac{2701}{15200} a^{3} - \frac{6681}{30400} a^{2} + \frac{87}{200} a + \frac{41}{100}$, $\frac{1}{189878810303206400} a^{9} + \frac{195996035819}{189878810303206400} a^{8} - \frac{17088363432363}{94939405151603200} a^{7} + \frac{6726538651727}{94939405151603200} a^{6} + \frac{459134636534201}{189878810303206400} a^{5} - \frac{10306273302937517}{189878810303206400} a^{4} - \frac{7142223987551337}{47469702575801600} a^{3} - \frac{11309686073976567}{47469702575801600} a^{2} - \frac{21472748652799}{78075168710200} a + \frac{37590557020173}{156150337420400}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}\times C_{10}\times C_{18040}$, which has order $7216000$ (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:  \( 135613.65526477623 \) (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

\(\Q(\sqrt{-95}) \), 5.5.5719140625.4

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ R ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$5$5.10.17.5$x^{10} - 5 x^{8} + 55$$10$$1$$17$$C_{10}$$[2]_{2}$
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$