↑

Properties

Label 15.1.36746829585...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 11^{12}\cdot 149^{5}$
Root discriminant $434.19$
Ramified primes $2, 3, 5, 11, 149$
Class number $75$ (GRH)
Class group $[5, 15]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![425268514, 183470980, -43337470, 102976665, -42451640, -2042470, 670115, 490815, -95580, 33455, 242, -1115, 285, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 5*x^13 + 285*x^12 - 1115*x^11 + 242*x^10 + 33455*x^9 - 95580*x^8 + 490815*x^7 + 670115*x^6 - 2042470*x^5 - 42451640*x^4 + 102976665*x^3 - 43337470*x^2 + 183470980*x + 425268514)
 
gp: K = bnfinit(x^15 - 5*x^14 + 5*x^13 + 285*x^12 - 1115*x^11 + 242*x^10 + 33455*x^9 - 95580*x^8 + 490815*x^7 + 670115*x^6 - 2042470*x^5 - 42451640*x^4 + 102976665*x^3 - 43337470*x^2 + 183470980*x + 425268514, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 5 x^{13} + 285 x^{12} - 1115 x^{11} + 242 x^{10} + 33455 x^{9} - 95580 x^{8} + 490815 x^{7} + 670115 x^{6} - 2042470 x^{5} - 42451640 x^{4} + 102976665 x^{3} - 43337470 x^{2} + 183470980 x + 425268514 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3674682958567667934420285323670000000000=-\,2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 11^{12}\cdot 149^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $434.19$
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:  $2, 3, 5, 11, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{15} a^{8} + \frac{1}{15} a^{7} - \frac{2}{15} a^{6} + \frac{2}{5} a^{4} + \frac{2}{15} a^{2} + \frac{2}{15} a - \frac{4}{15}$, $\frac{1}{15} a^{9} + \frac{2}{15} a^{7} + \frac{2}{15} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{15} a^{3} + \frac{4}{15} a + \frac{4}{15}$, $\frac{1}{135} a^{10} + \frac{4}{135} a^{8} + \frac{1}{15} a^{7} - \frac{8}{135} a^{6} - \frac{4}{15} a^{5} + \frac{44}{135} a^{4} + \frac{1}{3} a^{3} + \frac{23}{135} a^{2} - \frac{1}{5} a + \frac{17}{135}$, $\frac{1}{135} a^{11} + \frac{4}{135} a^{9} - \frac{17}{135} a^{7} - \frac{2}{15} a^{6} + \frac{44}{135} a^{5} - \frac{1}{15} a^{4} + \frac{23}{135} a^{3} - \frac{1}{3} a^{2} - \frac{1}{135} a + \frac{4}{15}$, $\frac{1}{675} a^{12} + \frac{1}{675} a^{11} - \frac{1}{675} a^{10} - \frac{1}{135} a^{9} - \frac{2}{135} a^{8} - \frac{26}{675} a^{7} - \frac{32}{225} a^{6} + \frac{26}{675} a^{5} + \frac{29}{135} a^{4} + \frac{28}{135} a^{3} + \frac{298}{675} a^{2} + \frac{278}{675} a + \frac{302}{675}$, $\frac{1}{2025} a^{13} - \frac{2}{2025} a^{11} - \frac{4}{2025} a^{10} - \frac{1}{405} a^{9} - \frac{16}{2025} a^{8} + \frac{31}{405} a^{7} - \frac{103}{2025} a^{6} + \frac{119}{2025} a^{5} - \frac{1}{405} a^{4} + \frac{158}{2025} a^{3} + \frac{131}{405} a^{2} - \frac{292}{675} a - \frac{77}{2025}$, $\frac{1}{1080650439513491612239558816065078265583903025} a^{14} - \frac{26543340564039251021424498724505224549171}{1080650439513491612239558816065078265583903025} a^{13} + \frac{774683241357780464742197532115396854356458}{1080650439513491612239558816065078265583903025} a^{12} + \frac{439287925248674785913508721277109180641908}{1080650439513491612239558816065078265583903025} a^{11} - \frac{2114034462107159528415625854542725679768896}{1080650439513491612239558816065078265583903025} a^{10} - \frac{33966857187784565830698040350885342909072716}{1080650439513491612239558816065078265583903025} a^{9} + \frac{1012114855813015478682737760419936016104107}{360216813171163870746519605355026088527967675} a^{8} - \frac{36800598582343898736778843743147498223666741}{360216813171163870746519605355026088527967675} a^{7} - \frac{7637223369143688346954277597822931262439486}{360216813171163870746519605355026088527967675} a^{6} - \frac{355261228864630275788660991938758041326065534}{1080650439513491612239558816065078265583903025} a^{5} - \frac{81048121266393070661271342536585268287503052}{1080650439513491612239558816065078265583903025} a^{4} + \frac{403849965611105333812218833263497146055967502}{1080650439513491612239558816065078265583903025} a^{3} + \frac{60921808279993449210047275490675173473389684}{1080650439513491612239558816065078265583903025} a^{2} - \frac{20648571662650054909615724416256906601292781}{1080650439513491612239558816065078265583903025} a - \frac{3393034778372740340676879220043674628290072}{9914224215720106534307879046468607941136725}$

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

Class group and class number

$C_{5}\times C_{15}$, which has order $75$ (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:  $7$
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:  \( 5559866715429.072 \) (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

3.1.8940.3, 5.1.148240125.2

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
$149$$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$