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Properties

Label 15.15.4635329218...1009.1
Degree $15$
Signature $[15, 0]$
Discriminant $7^{10}\cdot 71^{12}$
Root discriminant $110.77$
Ramified primes $7, 71$
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![-2911, -229827, 1025879, -1098512, -361390, 1178315, -421848, -244262, 183885, -14576, -13890, 2696, 330, -95, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 95*x^13 + 330*x^12 + 2696*x^11 - 13890*x^10 - 14576*x^9 + 183885*x^8 - 244262*x^7 - 421848*x^6 + 1178315*x^5 - 361390*x^4 - 1098512*x^3 + 1025879*x^2 - 229827*x - 2911)
 
gp: K = bnfinit(x^15 - 2*x^14 - 95*x^13 + 330*x^12 + 2696*x^11 - 13890*x^10 - 14576*x^9 + 183885*x^8 - 244262*x^7 - 421848*x^6 + 1178315*x^5 - 361390*x^4 - 1098512*x^3 + 1025879*x^2 - 229827*x - 2911, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 95 x^{13} + 330 x^{12} + 2696 x^{11} - 13890 x^{10} - 14576 x^{9} + 183885 x^{8} - 244262 x^{7} - 421848 x^{6} + 1178315 x^{5} - 361390 x^{4} - 1098512 x^{3} + 1025879 x^{2} - 229827 x - 2911 \)

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:  \(4635329218173515544706698901009=7^{10}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.77$
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:  $7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(497=7\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{497}(128,·)$, $\chi_{497}(1,·)$, $\chi_{497}(289,·)$, $\chi_{497}(72,·)$, $\chi_{497}(267,·)$, $\chi_{497}(270,·)$, $\chi_{497}(431,·)$, $\chi_{497}(338,·)$, $\chi_{497}(480,·)$, $\chi_{497}(214,·)$, $\chi_{497}(25,·)$, $\chi_{497}(57,·)$, $\chi_{497}(218,·)$, $\chi_{497}(380,·)$, $\chi_{497}(309,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} - \frac{2}{23} a^{11} + \frac{7}{23} a^{10} - \frac{8}{23} a^{9} - \frac{2}{23} a^{8} - \frac{4}{23} a^{7} - \frac{2}{23} a^{5} + \frac{5}{23} a^{4} - \frac{2}{23} a^{3} - \frac{2}{23} a^{2} + \frac{9}{23} a + \frac{9}{23}$, $\frac{1}{2231} a^{13} - \frac{3}{2231} a^{12} + \frac{423}{2231} a^{11} + \frac{399}{2231} a^{10} - \frac{339}{2231} a^{9} + \frac{941}{2231} a^{8} + \frac{1085}{2231} a^{7} + \frac{734}{2231} a^{6} - \frac{16}{2231} a^{5} + \frac{798}{2231} a^{4} + \frac{27}{97} a^{3} - \frac{1093}{2231} a^{2} - \frac{36}{97} a + \frac{152}{2231}$, $\frac{1}{9773876143794286568547361474621} a^{14} + \frac{1493535810392149490090861180}{9773876143794286568547361474621} a^{13} + \frac{42860681567312407901790870491}{9773876143794286568547361474621} a^{12} - \frac{2070514051060440430530355880672}{9773876143794286568547361474621} a^{11} + \frac{3378240799485296352660406399215}{9773876143794286568547361474621} a^{10} - \frac{678282369240850837308722322989}{9773876143794286568547361474621} a^{9} - \frac{2520232530920443658776887246196}{9773876143794286568547361474621} a^{8} - \frac{2927720494676342827296937356651}{9773876143794286568547361474621} a^{7} + \frac{1861484210169034993088960653460}{9773876143794286568547361474621} a^{6} - \frac{239748593697193922954338346787}{9773876143794286568547361474621} a^{5} + \frac{3800917528627857463458387208528}{9773876143794286568547361474621} a^{4} - \frac{22004921068831191188332038892}{238387223019372843135301499381} a^{3} + \frac{2973361993329891135858733763312}{9773876143794286568547361474621} a^{2} - \frac{1629865558524630237503063595033}{9773876143794286568547361474621} a + \frac{102292993759243645483155234811}{238387223019372843135301499381}$

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:  \( 26670515863.43193 \) (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

\(\Q(\zeta_{7})^+\), 5.5.25411681.1

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 $15$ $15$ $15$ R $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ $15$ $15$ $15$

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
$7$7.15.10.1$x^{15} + 4116 x^{6} - 2401 x^{3} + 1075648$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
$71$71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.1$x^{5} - 71$$5$$1$$4$$C_5$$[\ ]_{5}$