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Properties

Label 15.15.2172912230...3488.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 37^{5}\cdot 701^{4}\cdot 1061^{4}$
Root discriminant $194.60$
Ramified primes $2, 37, 701, 1061$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T32

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3936256, -9523200, 7688192, 17932800, -4059136, -10598656, 234224, 2128832, 73248, -184624, -9056, 7456, 303, -141, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 141*x^13 + 303*x^12 + 7456*x^11 - 9056*x^10 - 184624*x^9 + 73248*x^8 + 2128832*x^7 + 234224*x^6 - 10598656*x^5 - 4059136*x^4 + 17932800*x^3 + 7688192*x^2 - 9523200*x - 3936256)
 
gp: K = bnfinit(x^15 - 3*x^14 - 141*x^13 + 303*x^12 + 7456*x^11 - 9056*x^10 - 184624*x^9 + 73248*x^8 + 2128832*x^7 + 234224*x^6 - 10598656*x^5 - 4059136*x^4 + 17932800*x^3 + 7688192*x^2 - 9523200*x - 3936256, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 141 x^{13} + 303 x^{12} + 7456 x^{11} - 9056 x^{10} - 184624 x^{9} + 73248 x^{8} + 2128832 x^{7} + 234224 x^{6} - 10598656 x^{5} - 4059136 x^{4} + 17932800 x^{3} + 7688192 x^{2} - 9523200 x - 3936256 \)

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:  \(21729122307019732447767846220223488=2^{10}\cdot 37^{5}\cdot 701^{4}\cdot 1061^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $194.60$
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:  $2, 37, 701, 1061$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{7} + \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{9} - \frac{1}{32} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} + \frac{1}{64} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{512} a^{12} + \frac{1}{512} a^{11} + \frac{7}{512} a^{10} - \frac{5}{512} a^{9} + \frac{3}{128} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{8} a^{4} + \frac{7}{32} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1024} a^{13} - \frac{1}{1024} a^{12} + \frac{5}{1024} a^{11} - \frac{3}{1024} a^{10} - \frac{5}{512} a^{9} - \frac{1}{128} a^{8} - \frac{1}{32} a^{7} - \frac{1}{32} a^{6} - \frac{1}{64} a^{4} - \frac{1}{32} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{387812853451677178145899606016} a^{14} - \frac{69480595452719192599286467}{387812853451677178145899606016} a^{13} - \frac{159789928948866015380814045}{387812853451677178145899606016} a^{12} - \frac{2812917719599535877697142513}{387812853451677178145899606016} a^{11} + \frac{139152538089405939078187281}{12119151670364911817059362688} a^{10} - \frac{15061423401797499767533}{466121218090958146809975488} a^{9} - \frac{225488354987540806618477687}{12119151670364911817059362688} a^{8} - \frac{21620474331488930871201009}{12119151670364911817059362688} a^{7} - \frac{11529810782269667391087375}{195470188231692126081602624} a^{6} + \frac{369599844619271019414109023}{24238303340729823634118725376} a^{5} - \frac{171855285186677797964017907}{1514893958795613977132420336} a^{4} + \frac{203873414129287990829259437}{1514893958795613977132420336} a^{3} + \frac{380557366137668545974399479}{1514893958795613977132420336} a^{2} - \frac{120359290910770615936951835}{378723489698903494283105084} a - \frac{733091912049843113798513}{3054221691120189470025041}$

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:  \( 45564274779100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T32:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 750
The 65 conjugacy class representatives for [5^3]S(3) are not computed
Character table for [5^3]S(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 15 siblings: data not computed
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 $15$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ R $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$37$37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
701Data not computed
1061Data not computed