↑

Properties

Label 10.0.30815096784...1875.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,5^{5}\cdot 11^{9}\cdot 53^{5}$
Root discriminant $140.89$
Ramified primes $5, 11, 53$
Class number $156984$ (GRH)
Class group $[2, 78492]$ (GRH)
Galois group $C_{10}$ (as 10T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14841598399, -1065940063, 1065940063, -22329583, 22329583, -192391, 192391, -727, 727, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 727*x^8 - 727*x^7 + 192391*x^6 - 192391*x^5 + 22329583*x^4 - 22329583*x^3 + 1065940063*x^2 - 1065940063*x + 14841598399)
 
gp: K = bnfinit(x^10 - x^9 + 727*x^8 - 727*x^7 + 192391*x^6 - 192391*x^5 + 22329583*x^4 - 22329583*x^3 + 1065940063*x^2 - 1065940063*x + 14841598399, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 727 x^{8} - 727 x^{7} + 192391 x^{6} - 192391 x^{5} + 22329583 x^{4} - 22329583 x^{3} + 1065940063 x^{2} - 1065940063 x + 14841598399 \)

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:  \(-3081509678456114571875=-\,5^{5}\cdot 11^{9}\cdot 53^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $140.89$
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:  $5, 11, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2915=5\cdot 11\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{2915}(1,·)$, $\chi_{2915}(2914,·)$, $\chi_{2915}(1061,·)$, $\chi_{2915}(2119,·)$, $\chi_{2915}(2121,·)$, $\chi_{2915}(2384,·)$, $\chi_{2915}(531,·)$, $\chi_{2915}(794,·)$, $\chi_{2915}(796,·)$, $\chi_{2915}(1854,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{1547138539} a^{6} - \frac{116502422}{1547138539} a^{5} + \frac{396}{1547138539} a^{4} + \frac{232664215}{1547138539} a^{3} + \frac{39204}{1547138539} a^{2} - \frac{115547200}{1547138539} a + \frac{574992}{1547138539}$, $\frac{1}{1547138539} a^{7} + \frac{462}{1547138539} a^{5} - \frac{46532843}{1547138539} a^{4} + \frac{60984}{1547138539} a^{3} + \frac{92437760}{1547138539} a^{2} + \frac{2012472}{1547138539} a - \frac{43830998}{1547138539}$, $\frac{1}{1547138539} a^{8} - \frac{372262744}{1547138539} a^{5} - \frac{121968}{1547138539} a^{4} - \frac{645870379}{1547138539} a^{3} - \frac{16099776}{1547138539} a^{2} + \frac{736265076}{1547138539} a - \frac{265646304}{1547138539}$, $\frac{1}{1547138539} a^{9} - \frac{156816}{1547138539} a^{5} - \frac{207984960}{1547138539} a^{4} - \frac{27599616}{1547138539} a^{3} + \frac{767042465}{1547138539} a^{2} + \frac{522502795}{1547138539} a - \frac{64311141}{1547138539}$

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

Class group and class number

$C_{2}\times C_{78492}$, which has order $156984$ (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

\(\Q(\sqrt{-2915}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ R ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$53$53.10.5.2$x^{10} - 7890481 x^{2} + 1254586479$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$