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Properties

Label 10.0.9630096522760791.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 7^{5}\cdot 11^{9}$
Root discriminant $39.66$
Ramified primes $3, 7, 11$
Class number $132$
Class group $[2, 66]$
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![79531, -45156, 45156, -10781, 10781, -1156, 1156, -56, 56, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 56*x^8 - 56*x^7 + 1156*x^6 - 1156*x^5 + 10781*x^4 - 10781*x^3 + 45156*x^2 - 45156*x + 79531)
 
gp: K = bnfinit(x^10 - x^9 + 56*x^8 - 56*x^7 + 1156*x^6 - 1156*x^5 + 10781*x^4 - 10781*x^3 + 45156*x^2 - 45156*x + 79531, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 56 x^{8} - 56 x^{7} + 1156 x^{6} - 1156 x^{5} + 10781 x^{4} - 10781 x^{3} + 45156 x^{2} - 45156 x + 79531 \)

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:  \(-9630096522760791=-\,3^{5}\cdot 7^{5}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.66$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(230,·)$, $\chi_{231}(167,·)$, $\chi_{231}(41,·)$, $\chi_{231}(83,·)$, $\chi_{231}(148,·)$, $\chi_{231}(62,·)$, $\chi_{231}(169,·)$, $\chi_{231}(190,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{17621} a^{6} + \frac{1287}{17621} a^{5} + \frac{30}{17621} a^{4} - \frac{3067}{17621} a^{3} + \frac{225}{17621} a^{2} + \frac{2286}{17621} a + \frac{250}{17621}$, $\frac{1}{17621} a^{7} + \frac{35}{17621} a^{5} - \frac{6435}{17621} a^{4} + \frac{350}{17621} a^{3} - \frac{5353}{17621} a^{2} + \frac{875}{17621} a - \frac{4572}{17621}$, $\frac{1}{17621} a^{8} + \frac{1383}{17621} a^{5} - \frac{700}{17621} a^{4} - \frac{3734}{17621} a^{3} - \frac{7000}{17621} a^{2} + \frac{3523}{17621} a - \frac{8750}{17621}$, $\frac{1}{17621} a^{9} - \frac{900}{17621} a^{5} + \frac{7639}{17621} a^{4} + \frac{5621}{17621} a^{3} - \frac{8095}{17621} a^{2} + \frac{1492}{17621} a + \frac{6670}{17621}$

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

Class group and class number

$C_{2}\times C_{66}$, which has order $132$

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

\(\Q(\sqrt{-231}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3_7_11.2t1.1c1$1$ $ 3 \cdot 7 \cdot 11 $ $x^{2} - x + 58$ $C_2$ (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) $0$ $1$
* 1.3_7_11.10t1.1c1$1$ $ 3 \cdot 7 \cdot 11 $ $x^{10} - x^{9} + 56 x^{8} - 56 x^{7} + 1156 x^{6} - 1156 x^{5} + 10781 x^{4} - 10781 x^{3} + 45156 x^{2} - 45156 x + 79531$ $C_{10}$ (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) $0$ $1$
* 1.3_7_11.10t1.1c2$1$ $ 3 \cdot 7 \cdot 11 $ $x^{10} - x^{9} + 56 x^{8} - 56 x^{7} + 1156 x^{6} - 1156 x^{5} + 10781 x^{4} - 10781 x^{3} + 45156 x^{2} - 45156 x + 79531$ $C_{10}$ (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) $0$ $1$
* 1.3_7_11.10t1.1c3$1$ $ 3 \cdot 7 \cdot 11 $ $x^{10} - x^{9} + 56 x^{8} - 56 x^{7} + 1156 x^{6} - 1156 x^{5} + 10781 x^{4} - 10781 x^{3} + 45156 x^{2} - 45156 x + 79531$ $C_{10}$ (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) $0$ $1$
* 1.3_7_11.10t1.1c4$1$ $ 3 \cdot 7 \cdot 11 $ $x^{10} - x^{9} + 56 x^{8} - 56 x^{7} + 1156 x^{6} - 1156 x^{5} + 10781 x^{4} - 10781 x^{3} + 45156 x^{2} - 45156 x + 79531$ $C_{10}$ (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 *.