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Properties

Label 10.0.69066963272...2727.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,7^{5}\cdot 11^{8}\cdot 61^{8}$
Root discriminant $482.98$
Ramified primes $7, 11, 61$
Class number $2217025$ (GRH)
Class group $[5, 443405]$ (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![30499072, 5675680, 13955416, 6027876, 787334, -771947, 71277, 3626, -524, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 - 524*x^8 + 3626*x^7 + 71277*x^6 - 771947*x^5 + 787334*x^4 + 6027876*x^3 + 13955416*x^2 + 5675680*x + 30499072)
 
gp: K = bnfinit(x^10 - 3*x^9 - 524*x^8 + 3626*x^7 + 71277*x^6 - 771947*x^5 + 787334*x^4 + 6027876*x^3 + 13955416*x^2 + 5675680*x + 30499072, 1)
 

Normalized defining polynomial

\( x^{10} - 3 x^{9} - 524 x^{8} + 3626 x^{7} + 71277 x^{6} - 771947 x^{5} + 787334 x^{4} + 6027876 x^{3} + 13955416 x^{2} + 5675680 x + 30499072 \)

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:  \(-690669632728369005581442727=-\,7^{5}\cdot 11^{8}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $482.98$
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:  $7, 11, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4697=7\cdot 11\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4697}(1,·)$, $\chi_{4697}(20,·)$, $\chi_{4697}(302,·)$, $\chi_{4697}(400,·)$, $\chi_{4697}(1343,·)$, $\chi_{4697}(1644,·)$, $\chi_{4697}(1742,·)$, $\chi_{4697}(1961,·)$, $\chi_{4697}(3303,·)$, $\chi_{4697}(3375,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{32} a^{6} - \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{13696} a^{8} - \frac{3}{856} a^{7} - \frac{185}{6848} a^{6} + \frac{37}{3424} a^{5} + \frac{485}{13696} a^{4} + \frac{827}{3424} a^{3} - \frac{149}{3424} a^{2} - \frac{263}{856} a - \frac{47}{107}$, $\frac{1}{5554562030563038700544} a^{9} - \frac{7789595471080603}{2777281015281519350272} a^{8} + \frac{41457044464014560627}{2777281015281519350272} a^{7} - \frac{12861348492702345309}{694320253820379837568} a^{6} - \frac{344332500600876086715}{5554562030563038700544} a^{5} + \frac{100619913555658017163}{2777281015281519350272} a^{4} - \frac{331236586904156486779}{1388640507640759675136} a^{3} - \frac{153800143874993425523}{694320253820379837568} a^{2} + \frac{68786534437910325503}{173580063455094959392} a - \frac{4846908119794102537}{21697507931886869924}$

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

Class group and class number

$C_{5}\times C_{443405}$, which has order $2217025$ (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:  \( 400186.6439256542 \) (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

\(\Q(\sqrt{-7}) \), 5.5.202716958081.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$61$61.10.8.3$x^{10} + 183 x^{5} + 14884$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$