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Properties

Label 15.15.5902598614...1856.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{26}\cdot 3^{20}\cdot 7^{2}\cdot 11^{4}\cdot 181^{6}$
Root discriminant $282.75$
Ramified primes $2, 3, 7, 11, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T89

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3863552, -57125376, -56685888, 419212416, -51121392, -121663908, 8195912, 12590520, -441828, -624766, 10104, 16047, -84, -204, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 204*x^13 - 84*x^12 + 16047*x^11 + 10104*x^10 - 624766*x^9 - 441828*x^8 + 12590520*x^7 + 8195912*x^6 - 121663908*x^5 - 51121392*x^4 + 419212416*x^3 - 56685888*x^2 - 57125376*x + 3863552)
 
gp: K = bnfinit(x^15 - 204*x^13 - 84*x^12 + 16047*x^11 + 10104*x^10 - 624766*x^9 - 441828*x^8 + 12590520*x^7 + 8195912*x^6 - 121663908*x^5 - 51121392*x^4 + 419212416*x^3 - 56685888*x^2 - 57125376*x + 3863552, 1)
 

Normalized defining polynomial

\( x^{15} - 204 x^{13} - 84 x^{12} + 16047 x^{11} + 10104 x^{10} - 624766 x^{9} - 441828 x^{8} + 12590520 x^{7} + 8195912 x^{6} - 121663908 x^{5} - 51121392 x^{4} + 419212416 x^{3} - 56685888 x^{2} - 57125376 x + 3863552 \)

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:  \(5902598614225707628542494671273721856=2^{26}\cdot 3^{20}\cdot 7^{2}\cdot 11^{4}\cdot 181^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $282.75$
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:  $2, 3, 7, 11, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{4} a^{9} - \frac{1}{16} a^{8} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{448} a^{13} + \frac{5}{112} a^{11} + \frac{1}{16} a^{10} - \frac{81}{448} a^{9} + \frac{3}{56} a^{8} - \frac{15}{224} a^{7} - \frac{53}{112} a^{6} - \frac{9}{56} a^{5} + \frac{25}{56} a^{4} - \frac{25}{112} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{44494801444959969575398418327988024553069312} a^{14} - \frac{295589014112998999109377552650297451239}{1589100051605713199121372083142429448323904} a^{13} - \frac{295277143486051642672875652167497871798079}{11123700361239992393849604581997006138267328} a^{12} + \frac{95671959586842847958837770842458078180417}{1589100051605713199121372083142429448323904} a^{11} - \frac{3006673626183480624974374361526191186150209}{44494801444959969575398418327988024553069312} a^{10} + \frac{1841499087425067688621643039486545055857455}{11123700361239992393849604581997006138267328} a^{9} - \frac{2383433331936304976044857109271486779622679}{22247400722479984787699209163994012276534656} a^{8} + \frac{35270598238104847404364112188632189195487}{1011245487385453853986327689272455103478848} a^{7} + \frac{2101180072852940899293885627681308741236813}{5561850180619996196924802290998503069133664} a^{6} + \frac{236462298414395307450599855462397254759039}{505622743692726926993163844636227551739424} a^{5} - \frac{4058692888574145459693037394245824700017361}{11123700361239992393849604581997006138267328} a^{4} + \frac{3693681910872671135343672348215946871557}{28376786635816307127167358627543383005784} a^{3} - \frac{6683004965733610563033973335491378641191}{24829688306339268736271438799100460130061} a^{2} + \frac{1801962759928380882735010939700667469723}{9028977565941552267735068654218349138204} a - \frac{41786875882806023294317676817693109729}{322463484497912580990538166222083897793}$

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:  \( 609960897416000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T89:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 60 conjugacy class representatives for 1/2[S(3)^5]S(5) are not computed
Character table for 1/2[S(3)^5]S(5) is not computed

Intermediate fields

5.5.11531872.1

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
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 R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R R $15$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $15$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.10.8$x^{4} + 2 x^{2} + 3$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.6.11.4$x^{6} + 6 x^{4} + 4 x^{2} + 4 x + 14$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
$181$181.3.2.3$x^{3} - 724$$3$$1$$2$$C_3$$[\ ]_{3}$
181.6.0.1$x^{6} - x + 23$$1$$6$$0$$C_6$$[\ ]^{6}$
181.6.4.3$x^{6} + 6335 x^{3} + 10614564$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$