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Properties

Label 15.9.22057358695...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{24}\cdot 5^{20}\cdot 13^{10}$
Root discriminant $143.30$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T101

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79360, 310400, -120800, -778600, 132000, 244752, -201920, 19520, 33640, -7900, 1192, -590, -40, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 20*x^13 - 40*x^12 - 590*x^11 + 1192*x^10 - 7900*x^9 + 33640*x^8 + 19520*x^7 - 201920*x^6 + 244752*x^5 + 132000*x^4 - 778600*x^3 - 120800*x^2 + 310400*x + 79360)
 
gp: K = bnfinit(x^15 + 20*x^13 - 40*x^12 - 590*x^11 + 1192*x^10 - 7900*x^9 + 33640*x^8 + 19520*x^7 - 201920*x^6 + 244752*x^5 + 132000*x^4 - 778600*x^3 - 120800*x^2 + 310400*x + 79360, 1)
 

Normalized defining polynomial

\( x^{15} + 20 x^{13} - 40 x^{12} - 590 x^{11} + 1192 x^{10} - 7900 x^{9} + 33640 x^{8} + 19520 x^{7} - 201920 x^{6} + 244752 x^{5} + 132000 x^{4} - 778600 x^{3} - 120800 x^{2} + 310400 x + 79360 \)

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:  \(-220573586958400000000000000000000=-\,2^{24}\cdot 5^{20}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.30$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{40} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5}$, $\frac{1}{3200} a^{13} + \frac{3}{400} a^{12} + \frac{73}{800} a^{11} - \frac{41}{400} a^{10} + \frac{37}{320} a^{9} + \frac{23}{400} a^{8} - \frac{191}{800} a^{7} - \frac{79}{400} a^{6} - \frac{3}{50} a^{5} - \frac{11}{40} a^{3} - \frac{7}{20} a^{2} - \frac{9}{80} a - \frac{1}{20}$, $\frac{1}{858599325911128300032767704524800} a^{14} - \frac{11681817928340429144659186973}{214649831477782075008191926131200} a^{13} - \frac{1835847824394162141599266625303}{214649831477782075008191926131200} a^{12} + \frac{1584790806973437191629867659517}{21464983147778207500819192613120} a^{11} - \frac{2234881242435984905519867274231}{429299662955564150016383852262400} a^{10} + \frac{45747603294790439424598298813}{6707807233680689844005997691600} a^{9} + \frac{14994276753685201303622440368073}{214649831477782075008191926131200} a^{8} + \frac{8953786187852499726754637402339}{107324915738891037504095963065600} a^{7} - \frac{831007973277121806862181723281}{5366245786944551875204798153280} a^{6} - \frac{97362572593927516127209463457}{13415614467361379688011995383200} a^{5} - \frac{765773995447831795434662454691}{10732491573889103750409596306560} a^{4} - \frac{200598586479795365657482273369}{5366245786944551875204798153280} a^{3} - \frac{4834089791300476975061288969401}{21464983147778207500819192613120} a^{2} - \frac{196031980009534345925893509619}{536624578694455187520479815328} a + \frac{133948709735290415898542805979}{1341561446736137968801199538320}$

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:  \( 143671917417 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T101:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed
Character table for [S(5)^3]3=S(5)wr3 is not computed

Intermediate fields

3.3.169.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ $15$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.24.199$x^{12} - 8 x^{11} - 6 x^{8} + 8 x^{6} + 8 x^{5} + 16 x^{4} + 16 x^{3} - 8 x^{2} - 8$$4$$3$$24$12T92$[2, 2, 2, 3, 3, 3]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.5.9.3$x^{5} + 80$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.8.7$x^{5} + 10 x^{4} + 5$$5$$1$$8$$F_5$$[2]^{4}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$