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Properties

Label 15.11.1628711298...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{12}\cdot 5^{6}\cdot 7^{10}\cdot 127\cdot 167^{2}\cdot 1723^{4}\cdot 537221^{2}$
Root discriminant $1404.29$
Ramified primes $2, 5, 7, 127, 167, 1723, 537221$
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![-1723000, -18091500, -81928650, -208784525, -327507840, -324898985, -201456698, -73968941, -13717404, -289215, 324886, 39227, -1476, -369, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 369*x^13 - 1476*x^12 + 39227*x^11 + 324886*x^10 - 289215*x^9 - 13717404*x^8 - 73968941*x^7 - 201456698*x^6 - 324898985*x^5 - 327507840*x^4 - 208784525*x^3 - 81928650*x^2 - 18091500*x - 1723000)
 
gp: K = bnfinit(x^15 - 369*x^13 - 1476*x^12 + 39227*x^11 + 324886*x^10 - 289215*x^9 - 13717404*x^8 - 73968941*x^7 - 201456698*x^6 - 324898985*x^5 - 327507840*x^4 - 208784525*x^3 - 81928650*x^2 - 18091500*x - 1723000, 1)
 

Normalized defining polynomial

\( x^{15} - 369 x^{13} - 1476 x^{12} + 39227 x^{11} + 324886 x^{10} - 289215 x^{9} - 13717404 x^{8} - 73968941 x^{7} - 201456698 x^{6} - 324898985 x^{5} - 327507840 x^{4} - 208784525 x^{3} - 81928650 x^{2} - 18091500 x - 1723000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(162871129811887303525838290060177920845248000000=2^{12}\cdot 5^{6}\cdot 7^{10}\cdot 127\cdot 167^{2}\cdot 1723^{4}\cdot 537221^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1404.29$
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:  $2, 5, 7, 127, 167, 1723, 537221$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{8} - \frac{1}{25} a^{6} + \frac{1}{25} a^{5} - \frac{1}{25} a^{4} - \frac{8}{25} a^{3} + \frac{4}{25} a^{2} - \frac{2}{5}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{7} + \frac{1}{25} a^{6} - \frac{1}{25} a^{5} - \frac{8}{25} a^{4} + \frac{4}{25} a^{3} - \frac{2}{5} a$, $\frac{1}{25} a^{10} + \frac{1}{25} a^{7} - \frac{2}{25} a^{6} - \frac{2}{25} a^{5} + \frac{3}{25} a^{4} + \frac{2}{25} a^{3} + \frac{9}{25} a^{2} - \frac{2}{5}$, $\frac{1}{125} a^{11} + \frac{1}{125} a^{9} - \frac{1}{125} a^{8} - \frac{3}{125} a^{7} - \frac{4}{125} a^{6} + \frac{36}{125} a^{4} - \frac{61}{125} a^{3} - \frac{33}{125} a^{2} - \frac{4}{25} a + \frac{9}{25}$, $\frac{1}{625} a^{12} + \frac{11}{625} a^{10} - \frac{1}{625} a^{9} + \frac{7}{625} a^{8} - \frac{19}{625} a^{7} - \frac{11}{125} a^{6} + \frac{1}{625} a^{5} - \frac{291}{625} a^{4} - \frac{243}{625} a^{3} - \frac{13}{125} a^{2} + \frac{34}{125} a - \frac{3}{25}$, $\frac{1}{2500000} a^{13} - \frac{31}{125000} a^{12} - \frac{7429}{2500000} a^{11} + \frac{1213}{312500} a^{10} - \frac{20913}{2500000} a^{9} - \frac{18447}{1250000} a^{8} + \frac{17609}{500000} a^{7} + \frac{6873}{78125} a^{6} - \frac{224961}{2500000} a^{5} + \frac{86281}{1250000} a^{4} + \frac{249127}{500000} a^{3} - \frac{1103}{125000} a^{2} + \frac{4623}{100000} a + \frac{449}{50000}$, $\frac{1}{80000000} a^{14} + \frac{1}{8000000} a^{13} + \frac{9971}{80000000} a^{12} + \frac{74717}{40000000} a^{11} + \frac{980607}{80000000} a^{10} + \frac{174979}{20000000} a^{9} - \frac{28767}{3200000} a^{8} - \frac{311857}{40000000} a^{7} - \frac{3585281}{80000000} a^{6} + \frac{988783}{20000000} a^{5} + \frac{3810339}{16000000} a^{4} - \frac{2883601}{8000000} a^{3} + \frac{393511}{3200000} a^{2} - \frac{392053}{800000} a + \frac{29887}{160000}$

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:  $12$
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:  \( 4290598859040000000 \) (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

\(\Q(\zeta_{7})^+\)

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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$[\ ]^{3}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
127Data not computed
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.4.0.1$x^{4} - x + 60$$1$$4$$0$$C_4$$[\ ]^{4}$
167.4.2.2$x^{4} - 167 x^{2} + 139445$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
167.5.0.1$x^{5} - x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
1723Data not computed
537221Data not computed