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Properties

Label 10.0.80363589842...7291.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 11^{9}\cdot 107^{5}$
Root discriminant $155.06$
Ramified primes $3, 11, 107$
Class number $137456$ (GRH)
Class group $[22, 6248]$ (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![38337306481, -2292506481, 2292506481, -39706481, 39706481, -282481, 282481, -881, 881, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 881*x^8 - 881*x^7 + 282481*x^6 - 282481*x^5 + 39706481*x^4 - 39706481*x^3 + 2292506481*x^2 - 2292506481*x + 38337306481)
 
gp: K = bnfinit(x^10 - x^9 + 881*x^8 - 881*x^7 + 282481*x^6 - 282481*x^5 + 39706481*x^4 - 39706481*x^3 + 2292506481*x^2 - 2292506481*x + 38337306481, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 881 x^{8} - 881 x^{7} + 282481 x^{6} - 282481 x^{5} + 39706481 x^{4} - 39706481 x^{3} + 2292506481 x^{2} - 2292506481 x + 38337306481 \)

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:  \(-8036358984236448717291=-\,3^{5}\cdot 11^{9}\cdot 107^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.06$
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:  $3, 11, 107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3531=3\cdot 11\cdot 107\)
Dirichlet character group:    $\lbrace$$\chi_{3531}(1,·)$, $\chi_{3531}(322,·)$, $\chi_{3531}(643,·)$, $\chi_{3531}(1285,·)$, $\chi_{3531}(2246,·)$, $\chi_{3531}(2888,·)$, $\chi_{3531}(3209,·)$, $\chi_{3531}(3530,·)$, $\chi_{3531}(2248,·)$, $\chi_{3531}(1283,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3909299921} a^{6} - \frac{243117134}{3909299921} a^{5} + \frac{480}{3909299921} a^{4} + \frac{485644425}{3909299921} a^{3} + \frac{57600}{3909299921} a^{2} - \frac{241445210}{3909299921} a + \frac{1024000}{3909299921}$, $\frac{1}{3909299921} a^{7} + \frac{560}{3909299921} a^{5} - \frac{97128885}{3909299921} a^{4} + \frac{89600}{3909299921} a^{3} + \frac{193156168}{3909299921} a^{2} + \frac{3584000}{3909299921} a - \frac{92353122}{3909299921}$, $\frac{1}{3909299921} a^{8} - \frac{777031080}{3909299921} a^{5} - \frac{179200}{3909299921} a^{4} + \frac{1883272638}{3909299921} a^{3} - \frac{28672000}{3909299921} a^{2} - \frac{1708532757}{3909299921} a - \frac{573440000}{3909299921}$, $\frac{1}{3909299921} a^{9} - \frac{230400}{3909299921} a^{5} - \frac{434601378}{3909299921} a^{4} - \frac{49152000}{3909299921} a^{3} + \frac{1616179635}{3909299921} a^{2} + \frac{1697459921}{3909299921} a + \frac{466499265}{3909299921}$

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

Class group and class number

$C_{22}\times C_{6248}$, which has order $137456$ (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

\(\Q(\sqrt{-3531}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$107$107.10.5.2$x^{10} - 131079601 x^{2} + 126229655763$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$