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Properties

Label 10.0.58837612521...4375.3
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{5}\cdot 11^{9}\cdot 41^{8}$
Root discriminant $377.54$
Ramified primes $5, 11, 41$
Class number $1237220$ (GRH)
Class group $[1237220]$ (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![574971904, -121997088, 22742472, 2902660, -231118, -92907, 30309, 842, -288, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 - 288*x^8 + 842*x^7 + 30309*x^6 - 92907*x^5 - 231118*x^4 + 2902660*x^3 + 22742472*x^2 - 121997088*x + 574971904)
 
gp: K = bnfinit(x^10 - 3*x^9 - 288*x^8 + 842*x^7 + 30309*x^6 - 92907*x^5 - 231118*x^4 + 2902660*x^3 + 22742472*x^2 - 121997088*x + 574971904, 1)
 

Normalized defining polynomial

\( x^{10} - 3 x^{9} - 288 x^{8} + 842 x^{7} + 30309 x^{6} - 92907 x^{5} - 231118 x^{4} + 2902660 x^{3} + 22742472 x^{2} - 121997088 x + 574971904 \)

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:  \(-58837612521292212217534375=-\,5^{5}\cdot 11^{9}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $377.54$
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:  $5, 11, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2255=5\cdot 11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{2255}(256,·)$, $\chi_{2255}(1,·)$, $\chi_{2255}(1349,·)$, $\chi_{2255}(1289,·)$, $\chi_{2255}(141,·)$, $\chi_{2255}(16,·)$, $\chi_{2255}(1841,·)$, $\chi_{2255}(754,·)$, $\chi_{2255}(789,·)$, $\chi_{2255}(329,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{88} a^{5} + \frac{1}{22} a^{4} - \frac{9}{88} a^{3} - \frac{2}{11} a^{2} + \frac{5}{22} a - \frac{4}{11}$, $\frac{1}{352} a^{6} + \frac{1}{352} a^{5} + \frac{1}{352} a^{4} - \frac{3}{32} a^{3} - \frac{21}{176} a^{2} + \frac{5}{44} a + \frac{3}{11}$, $\frac{1}{9152} a^{7} - \frac{1}{9152} a^{6} + \frac{1}{832} a^{5} + \frac{365}{9152} a^{4} - \frac{21}{2288} a^{3} + \frac{19}{176} a^{2} - \frac{7}{143} a + \frac{1}{11}$, $\frac{1}{73216} a^{8} - \frac{1}{36608} a^{7} + \frac{3}{18304} a^{6} + \frac{21}{36608} a^{5} + \frac{1735}{73216} a^{4} + \frac{199}{9152} a^{3} - \frac{3401}{18304} a^{2} - \frac{1025}{4576} a$, $\frac{1}{22849464824111704064} a^{9} - \frac{33420063755425}{22849464824111704064} a^{8} - \frac{19570494637327}{878825570158142464} a^{7} - \frac{6333403691145981}{11424732412055852032} a^{6} + \frac{105785273604316161}{22849464824111704064} a^{5} + \frac{50506241465654991}{22849464824111704064} a^{4} + \frac{1100167830773728549}{5712366206027926016} a^{3} + \frac{1086115457984927187}{5712366206027926016} a^{2} + \frac{177425680826390775}{1428091551506981504} a + \frac{195094454672943}{858228095857561}$

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

Class group and class number

$C_{1237220}$, which has order $1237220$ (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:  \( 371368.7529642051 \) (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

\(\Q(\sqrt{-55}) \), 5.5.41371966801.4

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.6$x^{10} + 216513$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$41$41.10.8.3$x^{10} + 943 x^{5} + 242064$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$