↑

Properties

Label 15.15.2094858387...5200.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{20}\cdot 5^{2}\cdot 37^{5}\cdot 61^{3}\cdot 47221^{2}\cdot 47717^{2}$
Root discriminant $418.23$
Ramified primes $2, 5, 37, 61, 47221, 47717$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![839, -195583, -971575, 7818979, -6222041, -5944535, 3350873, 1588759, -417199, -171625, 18435, 8185, -243, -165, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 165*x^13 - 243*x^12 + 8185*x^11 + 18435*x^10 - 171625*x^9 - 417199*x^8 + 1588759*x^7 + 3350873*x^6 - 5944535*x^5 - 6222041*x^4 + 7818979*x^3 - 971575*x^2 - 195583*x + 839)
 
gp: K = bnfinit(x^15 - x^14 - 165*x^13 - 243*x^12 + 8185*x^11 + 18435*x^10 - 171625*x^9 - 417199*x^8 + 1588759*x^7 + 3350873*x^6 - 5944535*x^5 - 6222041*x^4 + 7818979*x^3 - 971575*x^2 - 195583*x + 839, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 165 x^{13} - 243 x^{12} + 8185 x^{11} + 18435 x^{10} - 171625 x^{9} - 417199 x^{8} + 1588759 x^{7} + 3350873 x^{6} - 5944535 x^{5} - 6222041 x^{4} + 7818979 x^{3} - 971575 x^{2} - 195583 x + 839 \)

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:  \(2094858387934572656078862668231684915200=2^{20}\cdot 5^{2}\cdot 37^{5}\cdot 61^{3}\cdot 47221^{2}\cdot 47717^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $418.23$
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:  $2, 5, 37, 61, 47221, 47717$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{32} a^{13} + \frac{1}{32} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{5}{32} a^{9} + \frac{1}{32} a^{8} + \frac{1}{8} a^{7} + \frac{3}{32} a^{5} - \frac{1}{32} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{32} a - \frac{1}{32}$, $\frac{1}{16734167708878324566904120937201408} a^{14} + \frac{6130299485726605823183185655567}{4183541927219581141726030234300352} a^{13} - \frac{814110934856246537457460463951769}{16734167708878324566904120937201408} a^{12} - \frac{163263374358588438986625203711645}{2091770963609790570863015117150176} a^{11} + \frac{559132486489299843544747264997041}{16734167708878324566904120937201408} a^{10} - \frac{241418215314305069193738263300525}{1045885481804895285431507558575088} a^{9} + \frac{2613514800602728412876143141894599}{16734167708878324566904120937201408} a^{8} + \frac{398508964836750837988194353522031}{4183541927219581141726030234300352} a^{7} - \frac{3958445676958966144975209429383325}{16734167708878324566904120937201408} a^{6} + \frac{126443287399232302000160910472591}{1045885481804895285431507558575088} a^{5} + \frac{3219572056044543165006573249146905}{16734167708878324566904120937201408} a^{4} - \frac{569516676895662430777214940694489}{4183541927219581141726030234300352} a^{3} + \frac{6434469812059350541641661262148239}{16734167708878324566904120937201408} a^{2} + \frac{437419960306267047707601962803831}{4183541927219581141726030234300352} a + \frac{2325064987241509857372512480571437}{16734167708878324566904120937201408}$

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:  \( 2900222328940000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(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 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 $15$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{3}^{2}$
2.12.18.66$x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{2} - 2$$12$$1$$18$12T98$[4/3, 4/3, 5/3, 5/3, 2]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
37Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.8.0.1$x^{8} - x + 17$$1$$8$$0$$C_8$$[\ ]^{8}$
47221Data not computed
47717Data not computed