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Properties

Label 15.3.20408319182...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{15}\cdot 5^{28}\cdot 73^{13}$
Root discriminant $1662.09$
Ramified primes $2, 5, 73$
Class number $180$ (GRH)
Class group $[6, 30]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-401691100, 244074250, -20459750, 68380625, -9127000, -2527180, 1279775, 377075, 124650, -34875, -4006, 1905, 265, -65, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 65*x^13 + 265*x^12 + 1905*x^11 - 4006*x^10 - 34875*x^9 + 124650*x^8 + 377075*x^7 + 1279775*x^6 - 2527180*x^5 - 9127000*x^4 + 68380625*x^3 - 20459750*x^2 + 244074250*x - 401691100)
 
gp: K = bnfinit(x^15 - 5*x^14 - 65*x^13 + 265*x^12 + 1905*x^11 - 4006*x^10 - 34875*x^9 + 124650*x^8 + 377075*x^7 + 1279775*x^6 - 2527180*x^5 - 9127000*x^4 + 68380625*x^3 - 20459750*x^2 + 244074250*x - 401691100, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 65 x^{13} + 265 x^{12} + 1905 x^{11} - 4006 x^{10} - 34875 x^{9} + 124650 x^{8} + 377075 x^{7} + 1279775 x^{6} - 2527180 x^{5} - 9127000 x^{4} + 68380625 x^{3} - 20459750 x^{2} + 244074250 x - 401691100 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2040831918205308698658487548828125000000000000000=2^{15}\cdot 5^{28}\cdot 73^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1662.09$
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:  $2, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3650} a^{10} + \frac{142}{365} a^{9} - \frac{123}{730} a^{8} - \frac{2}{5} a^{7} - \frac{152}{365} a^{6} - \frac{98}{1825} a^{5} - \frac{20}{73} a^{4} - \frac{17}{146} a^{2} + \frac{26}{73} a + \frac{31}{365}$, $\frac{1}{3650} a^{11} + \frac{287}{730} a^{9} - \frac{51}{365} a^{8} - \frac{152}{365} a^{7} + \frac{527}{1825} a^{6} - \frac{8}{365} a^{5} + \frac{3}{73} a^{4} - \frac{17}{146} a^{3} - \frac{22}{73} a^{2} + \frac{121}{365} a + \frac{29}{73}$, $\frac{1}{3650} a^{12} - \frac{151}{365} a^{9} + \frac{271}{730} a^{8} + \frac{527}{1825} a^{7} - \frac{158}{365} a^{6} + \frac{36}{365} a^{5} + \frac{5}{146} a^{4} - \frac{22}{73} a^{3} + \frac{307}{730} a^{2} + \frac{22}{73} a + \frac{9}{73}$, $\frac{1}{3650} a^{13} - \frac{129}{730} a^{9} - \frac{248}{1825} a^{8} - \frac{158}{365} a^{7} + \frac{101}{365} a^{6} - \frac{37}{730} a^{5} + \frac{307}{730} a^{3} + \frac{35}{73} a^{2} - \frac{5}{73} a + \frac{18}{73}$, $\frac{1}{713097476851753299901924816793341162250} a^{14} + \frac{6795162647154318098408084869483333}{71309747685175329990192481679334116225} a^{13} + \frac{1324489678977771224411928939290733}{28523899074070131996076992671733646490} a^{12} - \frac{14893292221692150526044773902751747}{142619495370350659980384963358668232450} a^{11} - \frac{1543467966409461297135823185611893}{28523899074070131996076992671733646490} a^{10} + \frac{140728172982915643901331962880111701679}{713097476851753299901924816793341162250} a^{9} - \frac{30710063416104636010959219134911970887}{142619495370350659980384963358668232450} a^{8} + \frac{12123950161884970677937403738808254931}{71309747685175329990192481679334116225} a^{7} + \frac{7182099904964004545482091764178401069}{28523899074070131996076992671733646490} a^{6} + \frac{11866989917241447161239522523759455508}{71309747685175329990192481679334116225} a^{5} - \frac{25127260699927175926235567196893229939}{71309747685175329990192481679334116225} a^{4} + \frac{13249770890471360858694690096271853389}{28523899074070131996076992671733646490} a^{3} - \frac{12487945059474239270843207976698087269}{28523899074070131996076992671733646490} a^{2} - \frac{849784343241378592591950108229040865}{2852389907407013199607699267173364649} a + \frac{1024020786257245103738006101751282004}{14261949537035065998038496335866823245}$

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

Class group and class number

$C_{6}\times C_{30}$, which has order $180$ (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:  $8$
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:  \( 55268394954893130 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.3.2920.1, 5.1.55465314453125.4

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

Sibling fields

Degree 30 siblings: 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.5$x^{10} + 55$$10$$1$$19$$F_5$$[9/4]_{4}$
$73$73.5.4.1$x^{5} - 73$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
73.10.9.2$x^{10} + 365$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$