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Properties

Label 15.5.11651852705...0000.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{12}\cdot 5^{6}\cdot 7^{2}\cdot 23^{9}\cdot 101^{4}\cdot 167\cdot 937^{2}\cdot 3677^{2}$
Root discriminant $1010.24$
Ramified primes $2, 5, 7, 23, 101, 167, 937, 3677$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2323000, 8130500, -8362800, -15970625, 34914690, -10495935, -35086868, 13736612, -316710, -1351089, 21856, 23904, -936, -216, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 216*x^13 - 936*x^12 + 23904*x^11 + 21856*x^10 - 1351089*x^9 - 316710*x^8 + 13736612*x^7 - 35086868*x^6 - 10495935*x^5 + 34914690*x^4 - 15970625*x^3 - 8362800*x^2 + 8130500*x - 2323000)
 
gp: K = bnfinit(x^15 - 216*x^13 - 936*x^12 + 23904*x^11 + 21856*x^10 - 1351089*x^9 - 316710*x^8 + 13736612*x^7 - 35086868*x^6 - 10495935*x^5 + 34914690*x^4 - 15970625*x^3 - 8362800*x^2 + 8130500*x - 2323000, 1)
 

Normalized defining polynomial

\( x^{15} - 216 x^{13} - 936 x^{12} + 23904 x^{11} + 21856 x^{10} - 1351089 x^{9} - 316710 x^{8} + 13736612 x^{7} - 35086868 x^{6} - 10495935 x^{5} + 34914690 x^{4} - 15970625 x^{3} - 8362800 x^{2} + 8130500 x - 2323000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1165185270593545081708137982665622113856000000=-\,2^{12}\cdot 5^{6}\cdot 7^{2}\cdot 23^{9}\cdot 101^{4}\cdot 167\cdot 937^{2}\cdot 3677^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1010.24$
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:  $2, 5, 7, 23, 101, 167, 937, 3677$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{9} + \frac{2}{5} a^{8} - \frac{1}{10} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} + \frac{1}{5} a^{9} + \frac{9}{20} a^{8} - \frac{1}{5} a^{7} + \frac{3}{10} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} + \frac{1}{10} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{100} a^{13} - \frac{1}{100} a^{11} + \frac{7}{50} a^{10} - \frac{11}{100} a^{9} + \frac{4}{25} a^{8} + \frac{23}{50} a^{7} - \frac{1}{5} a^{6} + \frac{1}{50} a^{5} + \frac{8}{25} a^{4} - \frac{1}{20} a^{3} - \frac{3}{10} a^{2}$, $\frac{1}{359129223273854557928928806657391327408439640344300} a^{14} - \frac{1151554899550634637333111424380500429383509088577}{359129223273854557928928806657391327408439640344300} a^{13} + \frac{1633192245219538131812833683430095407205068795527}{179564611636927278964464403328695663704219820172150} a^{12} - \frac{454026841628364116099241501744567999241555645707}{21125248427873797525231106273964195729908214137900} a^{11} - \frac{21122002152715032032805799494596315363008295054247}{179564611636927278964464403328695663704219820172150} a^{10} + \frac{32688336732317457236869391141995545878673009611843}{359129223273854557928928806657391327408439640344300} a^{9} + \frac{158830851793090418125505283706738599708921483305369}{359129223273854557928928806657391327408439640344300} a^{8} - \frac{8582646294132331958523479139923268657027428942318}{89782305818463639482232201664347831852109910086075} a^{7} + \frac{24787039109723107948329179899236410262045824342681}{179564611636927278964464403328695663704219820172150} a^{6} + \frac{31125474340416485989066026757532246955200399456109}{179564611636927278964464403328695663704219820172150} a^{5} + \frac{148747272392077249786750299097119929873000122762491}{359129223273854557928928806657391327408439640344300} a^{4} - \frac{1901949372262951095600931598111354373637935133781}{71825844654770911585785761331478265481687928068860} a^{3} + \frac{22403508679938300488985274462825072707735965187353}{71825844654770911585785761331478265481687928068860} a^{2} - \frac{1674146613137633414068836904889542401484352575787}{3591292232738545579289288066573913274084396403443} a - \frac{387630064932401048460129214663874033254417695017}{3591292232738545579289288066573913274084396403443}$

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:  $9$
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:  \( 149269191915000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed

Intermediate fields

3.1.23.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.10.0.1$x^{10} + 5 x^{2} - x + 5$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$23$23.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
23.10.9.2$x^{10} + 46$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
101.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
101.5.4.4$x^{5} + 808$$5$$1$$4$$C_5$$[\ ]_{5}$
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
167.4.0.1$x^{4} - x + 60$$1$$4$$0$$C_4$$[\ ]^{4}$
167.5.0.1$x^{5} - x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
937Data not computed
3677Data not computed