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Properties

Label 10.0.31732821028...9375.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 5^{17}\cdot 11^{8}\cdot 41^{8}$
Root discriminant $3549.37$
Ramified primes $3, 5, 11, 41$
Class number $364064800$ (GRH)
Class group $[2, 2, 2, 110, 413710]$ (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![9142215828856099, 140389724566575, 12256011666415, -163060479670, 706818475, -480905359, 2751100, -133045, 4510, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 4510*x^8 - 133045*x^7 + 2751100*x^6 - 480905359*x^5 + 706818475*x^4 - 163060479670*x^3 + 12256011666415*x^2 + 140389724566575*x + 9142215828856099)
 
gp: K = bnfinit(x^10 + 4510*x^8 - 133045*x^7 + 2751100*x^6 - 480905359*x^5 + 706818475*x^4 - 163060479670*x^3 + 12256011666415*x^2 + 140389724566575*x + 9142215828856099, 1)
 

Normalized defining polynomial

\( x^{10} + 4510 x^{8} - 133045 x^{7} + 2751100 x^{6} - 480905359 x^{5} + 706818475 x^{4} - 163060479670 x^{3} + 12256011666415 x^{2} + 140389724566575 x + 9142215828856099 \)

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:  \(-317328210286621261737856292724609375=-\,3^{5}\cdot 5^{17}\cdot 11^{8}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3549.37$
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:  $3, 5, 11, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(33825=3\cdot 5^{2}\cdot 11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{33825}(1,·)$, $\chi_{33825}(4321,·)$, $\chi_{33825}(4711,·)$, $\chi_{33825}(17339,·)$, $\chi_{33825}(19034,·)$, $\chi_{33825}(27406,·)$, $\chi_{33825}(30479,·)$, $\chi_{33825}(32924,·)$, $\chi_{33825}(33269,·)$, $\chi_{33825}(33466,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{4059} a^{5} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{12177} a^{6} - \frac{1}{12177} a^{5} - \frac{1}{27} a^{4} + \frac{2}{27} a^{3} - \frac{10}{27} a^{2} - \frac{1}{27} a + \frac{1}{27}$, $\frac{1}{36531} a^{7} + \frac{1}{36531} a^{5} + \frac{1}{81} a^{4} + \frac{7}{81} a^{3} - \frac{26}{81} a^{2} + \frac{1}{9} a + \frac{7}{81}$, $\frac{1}{35398539} a^{8} - \frac{386}{35398539} a^{7} + \frac{214}{35398539} a^{6} + \frac{1103}{35398539} a^{5} + \frac{3521}{78489} a^{4} + \frac{7535}{78489} a^{3} - \frac{5414}{78489} a^{2} + \frac{31321}{78489} a + \frac{94}{4131}$, $\frac{1}{51985010251058318999485101005191365339} a^{9} - \frac{497524890869168572004171710640}{51985010251058318999485101005191365339} a^{8} - \frac{73213475269381112261053597384433}{51985010251058318999485101005191365339} a^{7} - \frac{8336268539968772798124102738634}{277994707224910796788690379706905697} a^{6} - \frac{2827977904613279027308085337448930}{51985010251058318999485101005191365339} a^{5} + \frac{6042119947603080645401437252076564}{115266098117645940131896011097985289} a^{4} + \frac{14884114229236253725568931931593766}{115266098117645940131896011097985289} a^{3} - \frac{41003601539098440802697146898801270}{115266098117645940131896011097985289} a^{2} + \frac{38385802328301265834272898245979726}{115266098117645940131896011097985289} a - \frac{73729039918752227424494514941465}{224690249741999883298042906623753}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{110}\times C_{413710}$, which has order $364064800$ (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:  \( 1487469.5436229876 \) (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

\(\Q(\sqrt{-15}) \), 5.5.16160924531640625.6

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 R ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ R ${\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.10.17.27$x^{10} - 10 x^{8} + 10$$10$$1$$17$$C_{10}$$[2]_{2}$
$11$11.10.8.4$x^{10} - 781 x^{5} + 290521$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$41$41.10.8.3$x^{10} + 943 x^{5} + 242064$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$