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Properties

Label 15.15.2572344674...3521.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 31^{10}$
Root discriminant $67.20$
Ramified primes $11, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47081, -16054, -252062, 77338, 318268, -111611, -157405, 61557, 33507, -14085, -3182, 1422, 133, -63, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 63*x^13 + 133*x^12 + 1422*x^11 - 3182*x^10 - 14085*x^9 + 33507*x^8 + 61557*x^7 - 157405*x^6 - 111611*x^5 + 318268*x^4 + 77338*x^3 - 252062*x^2 - 16054*x + 47081)
 
gp: K = bnfinit(x^15 - 2*x^14 - 63*x^13 + 133*x^12 + 1422*x^11 - 3182*x^10 - 14085*x^9 + 33507*x^8 + 61557*x^7 - 157405*x^6 - 111611*x^5 + 318268*x^4 + 77338*x^3 - 252062*x^2 - 16054*x + 47081, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2572344674223769220522333521=11^{12}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.20$
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:  $11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(341=11\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{341}(1,·)$, $\chi_{341}(67,·)$, $\chi_{341}(36,·)$, $\chi_{341}(5,·)$, $\chi_{341}(335,·)$, $\chi_{341}(273,·)$, $\chi_{341}(180,·)$, $\chi_{341}(311,·)$, $\chi_{341}(56,·)$, $\chi_{341}(25,·)$, $\chi_{341}(218,·)$, $\chi_{341}(284,·)$, $\chi_{341}(125,·)$, $\chi_{341}(280,·)$, $\chi_{341}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{46} a^{12} + \frac{1}{23} a^{11} - \frac{2}{23} a^{10} - \frac{11}{46} a^{9} - \frac{11}{23} a^{8} + \frac{5}{23} a^{7} - \frac{5}{23} a^{6} - \frac{17}{46} a^{5} - \frac{17}{46} a^{4} - \frac{4}{23} a^{3} + \frac{3}{23} a^{2} - \frac{11}{23} a - \frac{1}{2}$, $\frac{1}{46} a^{13} - \frac{4}{23} a^{11} - \frac{3}{46} a^{10} + \frac{4}{23} a^{8} + \frac{8}{23} a^{7} + \frac{3}{46} a^{6} + \frac{17}{46} a^{5} - \frac{10}{23} a^{4} + \frac{11}{23} a^{3} + \frac{6}{23} a^{2} + \frac{21}{46} a$, $\frac{1}{21432297989233180085436105638} a^{14} - \frac{83418802314858398540884016}{10716148994616590042718052819} a^{13} + \frac{63642103611520365118509182}{10716148994616590042718052819} a^{12} - \frac{174289252130827635999253837}{10716148994616590042718052819} a^{11} - \frac{1305745576927150511900599}{98313293528592569199248191} a^{10} - \frac{10010604777878601218962948311}{21432297989233180085436105638} a^{9} + \frac{529392101276930714945154882}{10716148994616590042718052819} a^{8} - \frac{7282148003087547927797089629}{21432297989233180085436105638} a^{7} + \frac{5893744120179181982266270499}{21432297989233180085436105638} a^{6} - \frac{2615820834170913066190321709}{10716148994616590042718052819} a^{5} - \frac{974839268981658185376101563}{21432297989233180085436105638} a^{4} - \frac{2832312524621501228587448993}{21432297989233180085436105638} a^{3} - \frac{543556833727915521171257431}{10716148994616590042718052819} a^{2} - \frac{1942384847094288597932452883}{10716148994616590042718052819} a + \frac{529663975870120488188633}{10470101606855486118923354}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 920271857.6558672 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

3.3.961.1, \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ $15$ $15$ $15$ R $15$ $15$ $15$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ $15$

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
$11$11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
$31$31.15.10.1$x^{15} + 893730 x^{6} - 923521 x^{3} + 28629151000$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.31.3t1.1c1$1$ $ 31 $ $x^{3} - x^{2} - 10 x + 8$ $C_3$ (as 3T1) $0$ $1$
* 1.31.3t1.1c2$1$ $ 31 $ $x^{3} - x^{2} - 10 x + 8$ $C_3$ (as 3T1) $0$ $1$
* 1.11.5t1.1c1$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.11_31.15t1.1c1$1$ $ 11 \cdot 31 $ $x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081$ $C_{15}$ (as 15T1) $0$ $1$
* 1.11_31.15t1.1c2$1$ $ 11 \cdot 31 $ $x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081$ $C_{15}$ (as 15T1) $0$ $1$
* 1.11.5t1.1c2$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.11_31.15t1.1c3$1$ $ 11 \cdot 31 $ $x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081$ $C_{15}$ (as 15T1) $0$ $1$
* 1.11_31.15t1.1c4$1$ $ 11 \cdot 31 $ $x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081$ $C_{15}$ (as 15T1) $0$ $1$
* 1.11.5t1.1c3$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.11_31.15t1.1c5$1$ $ 11 \cdot 31 $ $x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081$ $C_{15}$ (as 15T1) $0$ $1$
* 1.11_31.15t1.1c6$1$ $ 11 \cdot 31 $ $x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081$ $C_{15}$ (as 15T1) $0$ $1$
* 1.11.5t1.1c4$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.11_31.15t1.1c7$1$ $ 11 \cdot 31 $ $x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081$ $C_{15}$ (as 15T1) $0$ $1$
* 1.11_31.15t1.1c8$1$ $ 11 \cdot 31 $ $x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081$ $C_{15}$ (as 15T1) $0$ $1$

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 *.