↑

Properties

Label 15.15.1515690136...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{4}\cdot 257^{5}\cdot 3617^{4}\cdot 9043^{2}\cdot 99103^{2}$
Root discriminant $6486.95$
Ramified primes $2, 5, 7, 257, 3617, 9043, 99103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T100

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-103706624000, -544459776000, -1232812492800, -1570831270400, -1232034693120, -611135258240, -189549759616, -34903903136, -3309446592, -64056600, 14049472, 853252, -13248, -1656, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1656*x^13 - 13248*x^12 + 853252*x^11 + 14049472*x^10 - 64056600*x^9 - 3309446592*x^8 - 34903903136*x^7 - 189549759616*x^6 - 611135258240*x^5 - 1232034693120*x^4 - 1570831270400*x^3 - 1232812492800*x^2 - 544459776000*x - 103706624000)
 
gp: K = bnfinit(x^15 - 1656*x^13 - 13248*x^12 + 853252*x^11 + 14049472*x^10 - 64056600*x^9 - 3309446592*x^8 - 34903903136*x^7 - 189549759616*x^6 - 611135258240*x^5 - 1232034693120*x^4 - 1570831270400*x^3 - 1232812492800*x^2 - 544459776000*x - 103706624000, 1)
 

Normalized defining polynomial

\( x^{15} - 1656 x^{13} - 13248 x^{12} + 853252 x^{11} + 14049472 x^{10} - 64056600 x^{9} - 3309446592 x^{8} - 34903903136 x^{7} - 189549759616 x^{6} - 611135258240 x^{5} - 1232034693120 x^{4} - 1570831270400 x^{3} - 1232812492800 x^{2} - 544459776000 x - 103706624000 \)

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:  \(1515690136126498500450210146631049938378469084270592000000=2^{18}\cdot 5^{6}\cdot 7^{4}\cdot 257^{5}\cdot 3617^{4}\cdot 9043^{2}\cdot 99103^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6486.95$
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:  $2, 5, 7, 257, 3617, 9043, 99103$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} + \frac{1}{20} a^{4} - \frac{1}{20} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{800} a^{8} - \frac{1}{200} a^{6} + \frac{1}{100} a^{5} + \frac{21}{200} a^{4} + \frac{9}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} + \frac{1}{200} a^{7} - \frac{1}{100} a^{6} - \frac{19}{800} a^{5} - \frac{3}{100} a^{4} + \frac{17}{400} a^{3} + \frac{1}{10} a^{2} - \frac{1}{20} a$, $\frac{1}{32000} a^{10} - \frac{1}{16000} a^{9} + \frac{1}{4000} a^{8} - \frac{9}{2000} a^{7} + \frac{17}{1600} a^{6} + \frac{59}{4000} a^{5} - \frac{163}{4000} a^{4} - \frac{129}{2000} a^{3} - \frac{99}{1000} a^{2} - \frac{19}{100} a + \frac{6}{25}$, $\frac{1}{128000} a^{11} + \frac{1}{32000} a^{9} + \frac{1}{4000} a^{8} + \frac{13}{32000} a^{7} - \frac{17}{2000} a^{6} - \frac{57}{3200} a^{5} + \frac{187}{2000} a^{4} - \frac{1}{500} a^{3} + \frac{123}{1000} a^{2} + \frac{43}{200} a + \frac{21}{50}$, $\frac{1}{1280000} a^{12} + \frac{1}{320000} a^{10} - \frac{1}{10000} a^{9} + \frac{13}{320000} a^{8} - \frac{107}{20000} a^{7} + \frac{231}{32000} a^{6} - \frac{23}{20000} a^{5} + \frac{459}{5000} a^{4} - \frac{347}{10000} a^{3} + \frac{363}{2000} a^{2} - \frac{219}{500} a + \frac{2}{5}$, $\frac{1}{5120000000} a^{13} - \frac{31}{128000000} a^{12} + \frac{1263}{640000000} a^{11} - \frac{161}{40000000} a^{10} - \frac{110047}{1280000000} a^{9} + \frac{57291}{160000000} a^{8} - \frac{670639}{128000000} a^{7} - \frac{482299}{40000000} a^{6} - \frac{1399233}{160000000} a^{5} + \frac{1820893}{40000000} a^{4} + \frac{625731}{8000000} a^{3} + \frac{8541}{1000000} a^{2} - \frac{182981}{400000} a + \frac{39797}{100000}$, $\frac{1}{1966080000000} a^{14} - \frac{11}{491520000000} a^{13} - \frac{49117}{245760000000} a^{12} - \frac{63007}{61440000000} a^{11} - \frac{3191839}{491520000000} a^{10} - \frac{4783157}{40960000000} a^{9} - \frac{18991017}{81920000000} a^{8} + \frac{19421833}{20480000000} a^{7} - \frac{460347649}{61440000000} a^{6} - \frac{36011629}{2560000000} a^{5} + \frac{1335576683}{15360000000} a^{4} + \frac{57310651}{768000000} a^{3} + \frac{65313167}{768000000} a^{2} - \frac{497761}{19200000} a - \frac{1006697}{9600000}$

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:  \( 1957834578960000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T100:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed
Character table for [1/2.S(5)^3]S(3) is not computed

Intermediate fields

3.3.257.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$[\ ]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
257Data not computed
3617Data not computed
9043Data not computed
99103Data not computed