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Properties

Label 15.9.50068967077...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{12}\cdot 3^{16}\cdot 5^{9}\cdot 23\cdot 43^{6}$
Root discriminant $81.90$
Ramified primes $2, 3, 5, 23, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T87

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, -28493, -5292, 21349, -39915, -1671, 32534, 10091, -18072, -250, 2412, -6, -89, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 17*x^13 - 89*x^12 - 6*x^11 + 2412*x^10 - 250*x^9 - 18072*x^8 + 10091*x^7 + 32534*x^6 - 1671*x^5 - 39915*x^4 + 21349*x^3 - 5292*x^2 - 28493*x + 43)
 
gp: K = bnfinit(x^15 - 17*x^13 - 89*x^12 - 6*x^11 + 2412*x^10 - 250*x^9 - 18072*x^8 + 10091*x^7 + 32534*x^6 - 1671*x^5 - 39915*x^4 + 21349*x^3 - 5292*x^2 - 28493*x + 43, 1)
 

Normalized defining polynomial

\( x^{15} - 17 x^{13} - 89 x^{12} - 6 x^{11} + 2412 x^{10} - 250 x^{9} - 18072 x^{8} + 10091 x^{7} + 32534 x^{6} - 1671 x^{5} - 39915 x^{4} + 21349 x^{3} - 5292 x^{2} - 28493 x + 43 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-50068967077842268536000000000=-\,2^{12}\cdot 3^{16}\cdot 5^{9}\cdot 23\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.90$
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:  $2, 3, 5, 23, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{10} + \frac{2}{15} a^{9} + \frac{4}{15} a^{8} - \frac{7}{15} a^{6} + \frac{1}{5} a^{5} + \frac{1}{15} a^{4} - \frac{1}{15} a^{3} - \frac{2}{15} a^{2} - \frac{1}{15}$, $\frac{1}{15} a^{11} + \frac{7}{15} a^{8} - \frac{7}{15} a^{7} + \frac{2}{15} a^{6} - \frac{1}{3} a^{5} - \frac{1}{5} a^{4} + \frac{4}{15} a^{2} - \frac{1}{15} a + \frac{2}{15}$, $\frac{1}{15} a^{12} + \frac{2}{15} a^{9} - \frac{7}{15} a^{8} - \frac{1}{5} a^{7} + \frac{1}{3} a^{6} - \frac{1}{5} a^{5} - \frac{1}{15} a^{3} - \frac{1}{15} a^{2} - \frac{1}{5} a - \frac{1}{3}$, $\frac{1}{1275} a^{13} + \frac{7}{1275} a^{12} + \frac{12}{425} a^{11} + \frac{7}{425} a^{10} - \frac{28}{255} a^{9} - \frac{118}{425} a^{8} + \frac{94}{425} a^{7} - \frac{4}{1275} a^{6} - \frac{208}{425} a^{5} + \frac{1}{85} a^{4} + \frac{613}{1275} a^{3} - \frac{9}{25} a^{2} - \frac{322}{1275} a + \frac{478}{1275}$, $\frac{1}{7578015149893451291933912475} a^{14} - \frac{1913616254757221593697752}{7578015149893451291933912475} a^{13} - \frac{76432433945675831227932697}{7578015149893451291933912475} a^{12} + \frac{135768559056446005569136222}{7578015149893451291933912475} a^{11} - \frac{89696871826412324023393864}{7578015149893451291933912475} a^{10} + \frac{17347637060941258079697082}{2526005049964483763977970825} a^{9} - \frac{1825954904976575899139514257}{7578015149893451291933912475} a^{8} + \frac{3259855311915277034803794593}{7578015149893451291933912475} a^{7} + \frac{382273091678747746954118844}{2526005049964483763977970825} a^{6} + \frac{1789059993219744516375210736}{7578015149893451291933912475} a^{5} + \frac{5546059582707953121522079}{445765597052555958349053675} a^{4} - \frac{866927530962946222358825257}{2526005049964483763977970825} a^{3} - \frac{359613571871972894031853892}{2526005049964483763977970825} a^{2} + \frac{48165030600315907905910599}{194308080766498751075228525} a - \frac{558930733155910265731748567}{7578015149893451291933912475}$

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:  $11$
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:  \( 3081208230.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T87:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 155520
The 63 conjugacy class representatives for [S(3)^5]F(5)=S(3)wrF(5) are not computed
Character table for [S(3)^5]F(5)=S(3)wrF(5) is not computed

Intermediate fields

5.5.3698000.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ R ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.12.16.34$x^{12} + 48 x^{11} + 108 x^{10} + 111 x^{9} - 72 x^{8} - 54 x^{7} + 72 x^{6} + 108 x^{5} + 81 x^{4} + 54 x^{3} - 81 x^{2} - 81$$3$$4$$16$12T173$[2, 2, 2, 2]^{8}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.0.1$x^{8} + x^{2} - 2 x + 5$$1$$8$$0$$C_8$$[\ ]^{8}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.12.6.2$x^{12} - 147008443 x^{2} + 164355439274$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$