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Properties

Label 15.15.1831650872...1648.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{15}\cdot 11^{13}\cdot 61^{13}$
Root discriminant $563.45$
Ramified primes $2, 11, 61$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $S_3 \times C_5$ (as 15T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34747476224, 132200896640, 113736612400, -10574547144, -35037744764, -3528834550, 3071208794, 443602723, -81228726, -13426470, 777118, 154340, -2038, -676, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 676*x^13 - 2038*x^12 + 154340*x^11 + 777118*x^10 - 13426470*x^9 - 81228726*x^8 + 443602723*x^7 + 3071208794*x^6 - 3528834550*x^5 - 35037744764*x^4 - 10574547144*x^3 + 113736612400*x^2 + 132200896640*x + 34747476224)
 
gp: K = bnfinit(x^15 - 676*x^13 - 2038*x^12 + 154340*x^11 + 777118*x^10 - 13426470*x^9 - 81228726*x^8 + 443602723*x^7 + 3071208794*x^6 - 3528834550*x^5 - 35037744764*x^4 - 10574547144*x^3 + 113736612400*x^2 + 132200896640*x + 34747476224, 1)
 

Normalized defining polynomial

\( x^{15} - 676 x^{13} - 2038 x^{12} + 154340 x^{11} + 777118 x^{10} - 13426470 x^{9} - 81228726 x^{8} + 443602723 x^{7} + 3071208794 x^{6} - 3528834550 x^{5} - 35037744764 x^{4} - 10574547144 x^{3} + 113736612400 x^{2} + 132200896640 x + 34747476224 \)

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:  \(183165087243372408306736095370388796571648=2^{15}\cdot 11^{13}\cdot 61^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $563.45$
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:  $2, 11, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{8} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{9} - \frac{1}{8} a^{6} + \frac{1}{32} a^{5} - \frac{1}{8} a^{4} + \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{352} a^{10} + \frac{1}{352} a^{9} + \frac{5}{88} a^{7} + \frac{13}{352} a^{6} + \frac{37}{352} a^{5} + \frac{25}{176} a^{4} - \frac{9}{176} a^{3} - \frac{5}{44} a - \frac{1}{11}$, $\frac{1}{704} a^{11} - \frac{1}{704} a^{10} - \frac{1}{352} a^{9} + \frac{5}{176} a^{8} - \frac{27}{704} a^{7} - \frac{7}{64} a^{6} - \frac{3}{88} a^{5} + \frac{73}{352} a^{4} + \frac{9}{176} a^{3} + \frac{17}{88} a^{2} + \frac{3}{44} a + \frac{1}{11}$, $\frac{1}{297088} a^{12} - \frac{27}{74272} a^{11} - \frac{191}{297088} a^{10} - \frac{823}{148544} a^{9} - \frac{1573}{27008} a^{8} + \frac{3227}{74272} a^{7} + \frac{2253}{27008} a^{6} - \frac{511}{148544} a^{5} + \frac{33735}{148544} a^{4} + \frac{17607}{74272} a^{3} - \frac{1769}{37136} a^{2} - \frac{5627}{18568} a - \frac{1257}{4642}$, $\frac{1}{594176} a^{13} + \frac{383}{594176} a^{11} - \frac{47}{37136} a^{10} - \frac{107}{594176} a^{9} - \frac{111}{4642} a^{8} + \frac{18229}{594176} a^{7} - \frac{1183}{37136} a^{6} + \frac{10619}{297088} a^{5} - \frac{46}{2321} a^{4} + \frac{2251}{18568} a^{3} - \frac{7849}{18568} a^{2} + \frac{41}{88} a - \frac{569}{4642}$, $\frac{1}{131503099926884969868233529702084204620060294707311616} a^{14} + \frac{55084622179913705298119938590733119912691646451}{131503099926884969868233529702084204620060294707311616} a^{13} - \frac{73584516421965677350439105027560119775707765843}{131503099926884969868233529702084204620060294707311616} a^{12} - \frac{87820662014462325626729799170109185238478757699023}{131503099926884969868233529702084204620060294707311616} a^{11} + \frac{927651197871187832607426373887644042089423761043}{667528426024796801361591521330376673198275607651328} a^{10} + \frac{1606293331644139764588105157326888785980458069481179}{131503099926884969868233529702084204620060294707311616} a^{9} + \frac{4614534190840458124496637912048992667432501831458403}{131503099926884969868233529702084204620060294707311616} a^{8} - \frac{6583263264112749869093145514157913811419605099252045}{131503099926884969868233529702084204620060294707311616} a^{7} - \frac{4017470416905356067152500717198090072777487265232051}{32875774981721242467058382425521051155015073676827904} a^{6} - \frac{4546630346298666450579155793635755513174726897629165}{65751549963442484934116764851042102310030147353655808} a^{5} + \frac{5757008030076114655349443729338106253008388064430763}{32875774981721242467058382425521051155015073676827904} a^{4} + \frac{592056598891529351724633265205833909454127135303285}{16437887490860621233529191212760525577507536838413952} a^{3} - \frac{2720582439693661637375436896373751764574548029315825}{8218943745430310616764595606380262788753768419206976} a^{2} + \frac{265070825345578334748312354723074920951047682275303}{1027367968178788827095574450797532848594221052400872} a - \frac{255205410019560013083054175240264948602035602984923}{513683984089394413547787225398766424297110526200436}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 784994314915144300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times S_3$ (as 15T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 15 conjugacy class representatives for $S_3 \times C_5$
Character table for $S_3 \times C_5$

Intermediate fields

3.3.5368.1, 5.5.202716958081.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $15$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R $15$ $15$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$61$61.5.4.2$x^{5} + 122$$5$$1$$4$$C_5$$[\ ]_{5}$
61.10.9.4$x^{10} - 3904$$10$$1$$9$$C_{10}$$[\ ]_{10}$