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Properties

Label 10.0.685948419200000.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{10}\cdot 5^{5}\cdot 11^{8}$
Root discriminant $30.45$
Ramified primes $2, 5, 11$
Class number $62$
Class group $[62]$
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![13201, -2936, 6258, -1204, 1447, -242, 206, -26, 18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 18*x^8 - 26*x^7 + 206*x^6 - 242*x^5 + 1447*x^4 - 1204*x^3 + 6258*x^2 - 2936*x + 13201)
 
gp: K = bnfinit(x^10 - 2*x^9 + 18*x^8 - 26*x^7 + 206*x^6 - 242*x^5 + 1447*x^4 - 1204*x^3 + 6258*x^2 - 2936*x + 13201, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} + 18 x^{8} - 26 x^{7} + 206 x^{6} - 242 x^{5} + 1447 x^{4} - 1204 x^{3} + 6258 x^{2} - 2936 x + 13201 \)

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:  \(-685948419200000=-\,2^{10}\cdot 5^{5}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.45$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(220=2^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(199,·)$, $\chi_{220}(201,·)$, $\chi_{220}(141,·)$, $\chi_{220}(81,·)$, $\chi_{220}(179,·)$, $\chi_{220}(181,·)$, $\chi_{220}(119,·)$, $\chi_{220}(59,·)$, $\chi_{220}(159,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{66083891831981} a^{9} - \frac{8873359471445}{66083891831981} a^{8} - \frac{10078509859892}{66083891831981} a^{7} + \frac{744874712032}{1536834693767} a^{6} + \frac{7781897825066}{66083891831981} a^{5} + \frac{25422666925077}{66083891831981} a^{4} - \frac{26347616163684}{66083891831981} a^{3} + \frac{1930641386926}{66083891831981} a^{2} - \frac{4467912508743}{66083891831981} a + \frac{290047441749}{1536834693767}$

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

Class group and class number

$C_{62}$, which has order $62$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 26.1711060094 \)
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

\(\Q(\sqrt{-5}) \), \(\Q(\zeta_{11})^+\)

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2_5.2t1.1c1$1$ $ 2^{2} \cdot 5 $ $x^{2} + 5$ $C_2$ (as 2T1) $1$ $-1$
* 1.11.5t1.1c1$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.2e2_5_11.10t1.1c1$1$ $ 2^{2} \cdot 5 \cdot 11 $ $x^{10} - 2 x^{9} + 18 x^{8} - 26 x^{7} + 206 x^{6} - 242 x^{5} + 1447 x^{4} - 1204 x^{3} + 6258 x^{2} - 2936 x + 13201$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.1c2$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.2e2_5_11.10t1.1c2$1$ $ 2^{2} \cdot 5 \cdot 11 $ $x^{10} - 2 x^{9} + 18 x^{8} - 26 x^{7} + 206 x^{6} - 242 x^{5} + 1447 x^{4} - 1204 x^{3} + 6258 x^{2} - 2936 x + 13201$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.1c3$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.2e2_5_11.10t1.1c3$1$ $ 2^{2} \cdot 5 \cdot 11 $ $x^{10} - 2 x^{9} + 18 x^{8} - 26 x^{7} + 206 x^{6} - 242 x^{5} + 1447 x^{4} - 1204 x^{3} + 6258 x^{2} - 2936 x + 13201$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.1c4$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.2e2_5_11.10t1.1c4$1$ $ 2^{2} \cdot 5 \cdot 11 $ $x^{10} - 2 x^{9} + 18 x^{8} - 26 x^{7} + 206 x^{6} - 242 x^{5} + 1447 x^{4} - 1204 x^{3} + 6258 x^{2} - 2936 x + 13201$ $C_{10}$ (as 10T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.