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Properties

Label 15.11.2968548715...9392.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{10}\cdot 37^{5}\cdot 151^{2}\cdot 44269^{2}\cdot 305873^{2}$
Root discriminant $231.65$
Ramified primes $2, 37, 151, 44269, 305873$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T60

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -76800, -167680, 1038400, 301216, -728448, -121800, 38320, 2562, 15805, 1753, -1100, -79, -28, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 28*x^13 - 79*x^12 - 1100*x^11 + 1753*x^10 + 15805*x^9 + 2562*x^8 + 38320*x^7 - 121800*x^6 - 728448*x^5 + 301216*x^4 + 1038400*x^3 - 167680*x^2 - 76800*x + 4096)
 
gp: K = bnfinit(x^15 - x^14 - 28*x^13 - 79*x^12 - 1100*x^11 + 1753*x^10 + 15805*x^9 + 2562*x^8 + 38320*x^7 - 121800*x^6 - 728448*x^5 + 301216*x^4 + 1038400*x^3 - 167680*x^2 - 76800*x + 4096, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 28 x^{13} - 79 x^{12} - 1100 x^{11} + 1753 x^{10} + 15805 x^{9} + 2562 x^{8} + 38320 x^{7} - 121800 x^{6} - 728448 x^{5} + 301216 x^{4} + 1038400 x^{3} - 167680 x^{2} - 76800 x + 4096 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(296854871574412530032892056987579392=2^{10}\cdot 37^{5}\cdot 151^{2}\cdot 44269^{2}\cdot 305873^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $231.65$
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:  $2, 37, 151, 44269, 305873$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} + \frac{1}{6} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{24} a^{4} + \frac{5}{24} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3}$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{9} + \frac{1}{12} a^{8} + \frac{3}{16} a^{7} + \frac{1}{4} a^{6} - \frac{23}{48} a^{5} + \frac{5}{48} a^{4} - \frac{7}{24} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{96} a^{11} - \frac{1}{96} a^{10} - \frac{11}{96} a^{8} + \frac{5}{24} a^{7} + \frac{37}{96} a^{6} - \frac{19}{96} a^{5} - \frac{3}{16} a^{4} + \frac{7}{24} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{192} a^{12} - \frac{1}{192} a^{11} - \frac{1}{64} a^{9} + \frac{1}{16} a^{8} - \frac{9}{64} a^{7} + \frac{53}{192} a^{6} - \frac{3}{32} a^{5} + \frac{3}{16} a^{4} + \frac{1}{24} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{377088} a^{13} - \frac{665}{377088} a^{12} + \frac{91}{94272} a^{11} - \frac{3839}{377088} a^{10} - \frac{1465}{94272} a^{9} - \frac{34871}{377088} a^{8} - \frac{13051}{377088} a^{7} - \frac{70771}{188544} a^{6} - \frac{7415}{23568} a^{5} + \frac{8963}{47136} a^{4} - \frac{439}{11784} a^{3} + \frac{5125}{11784} a^{2} - \frac{1753}{5892} a + \frac{440}{1473}$, $\frac{1}{639834965658239429481476711424} a^{14} - \frac{466934022113022320159869}{639834965658239429481476711424} a^{13} + \frac{19486220540407073336994547}{39989685353639964342592294464} a^{12} + \frac{54301627185155524439508049}{639834965658239429481476711424} a^{11} + \frac{633088282061337692247585011}{79979370707279928685184588928} a^{10} + \frac{4394376530884953211036496659}{213278321886079809827158903808} a^{9} + \frac{45375614523851492478642312001}{639834965658239429481476711424} a^{8} - \frac{77245294085923075097949789373}{319917482829119714740738355712} a^{7} - \frac{31982631463001215866789997675}{79979370707279928685184588928} a^{6} - \frac{3738966879616492891260176947}{26659790235759976228394862976} a^{5} + \frac{2756351879927625677685783583}{19994842676819982171296147232} a^{4} + \frac{6206713470537056159382741481}{19994842676819982171296147232} a^{3} + \frac{2755085517286141061398119203}{9997421338409991085648073616} a^{2} + \frac{79062963252932738746566415}{624838833650624442853004601} a - \frac{73665467209585933799953071}{208279611216874814284334867}$

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:  $12$
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:  \( 10419229882500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T60:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6000
The 40 conjugacy class representatives for [D(5)^3]S(3)=D(5)wrS(3)
Character table for [D(5)^3]S(3)=D(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
37Data not computed
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
44269Data not computed
305873Data not computed