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Properties

Label 15.5.45574341341...9296.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{8}\cdot 17^{5}\cdot 31^{8}\cdot 43^{5}$
Root discriminant $81.39$
Ramified primes $2, 17, 31, 43$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 15T51

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-710752, -86972, -35034, -403447, 6957, 48450, -60845, 13460, 20521, -4208, -2579, 496, 159, -30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 30*x^13 + 159*x^12 + 496*x^11 - 2579*x^10 - 4208*x^9 + 20521*x^8 + 13460*x^7 - 60845*x^6 + 48450*x^5 + 6957*x^4 - 403447*x^3 - 35034*x^2 - 86972*x - 710752)
 
gp: K = bnfinit(x^15 - 5*x^14 - 30*x^13 + 159*x^12 + 496*x^11 - 2579*x^10 - 4208*x^9 + 20521*x^8 + 13460*x^7 - 60845*x^6 + 48450*x^5 + 6957*x^4 - 403447*x^3 - 35034*x^2 - 86972*x - 710752, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 30 x^{13} + 159 x^{12} + 496 x^{11} - 2579 x^{10} - 4208 x^{9} + 20521 x^{8} + 13460 x^{7} - 60845 x^{6} + 48450 x^{5} + 6957 x^{4} - 403447 x^{3} - 35034 x^{2} - 86972 x - 710752 \)

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:  \(-45574341341445246456603159296=-\,2^{8}\cdot 17^{5}\cdot 31^{8}\cdot 43^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.39$
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:  $2, 17, 31, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{14} a^{10} + \frac{1}{14} a^{9} + \frac{1}{14} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{5}{14} a^{4} - \frac{5}{14} a^{3} + \frac{5}{14} a^{2} - \frac{3}{7} a$, $\frac{1}{28} a^{11} - \frac{1}{28} a^{10} + \frac{3}{14} a^{9} + \frac{5}{28} a^{7} - \frac{3}{28} a^{6} - \frac{1}{7} a^{5} + \frac{3}{14} a^{4} - \frac{13}{28} a^{3} + \frac{5}{28} a^{2} + \frac{3}{7} a$, $\frac{1}{196} a^{12} - \frac{3}{196} a^{11} + \frac{2}{49} a^{9} + \frac{39}{196} a^{8} - \frac{29}{196} a^{7} - \frac{3}{14} a^{6} + \frac{45}{98} a^{5} - \frac{93}{196} a^{4} + \frac{43}{196} a^{3} + \frac{8}{49} a^{2} - \frac{22}{49} a - \frac{2}{7}$, $\frac{1}{1372} a^{13} + \frac{1}{1372} a^{12} + \frac{4}{343} a^{11} - \frac{12}{343} a^{10} - \frac{181}{1372} a^{9} - \frac{97}{1372} a^{8} + \frac{159}{686} a^{7} - \frac{79}{343} a^{6} + \frac{617}{1372} a^{5} + \frac{27}{196} a^{4} - \frac{255}{686} a^{3} + \frac{69}{686} a^{2} + \frac{293}{686} a - \frac{22}{49}$, $\frac{1}{10199872613190683613798770092} a^{14} + \frac{308603921517578657695014}{2549968153297670903449692523} a^{13} - \frac{9877411177397138260174783}{5099936306595341806899385046} a^{12} + \frac{34343809776720292456202317}{10199872613190683613798770092} a^{11} + \frac{226155026737016970022395341}{10199872613190683613798770092} a^{10} + \frac{47866593568248510078452635}{2549968153297670903449692523} a^{9} - \frac{16152222281046926327704238}{364281164756810129064241789} a^{8} - \frac{231874136855743840657511063}{10199872613190683613798770092} a^{7} - \frac{946157772035750592403783379}{10199872613190683613798770092} a^{6} + \frac{1243501581466384966924597415}{5099936306595341806899385046} a^{5} + \frac{2320992594904189048733729857}{5099936306595341806899385046} a^{4} - \frac{58276465185206134318995289}{10199872613190683613798770092} a^{3} + \frac{1855375744674087251608569929}{5099936306595341806899385046} a^{2} + \frac{1342677176337724268962352603}{5099936306595341806899385046} a - \frac{7224695236514986376986277}{364281164756810129064241789}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1950706707.66 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T51:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 38 conjugacy class representatives for [1/2.D(5)^3]S(3)
Character table for [1/2.D(5)^3]S(3) is not computed

Intermediate fields

3.1.731.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 $15$ $15$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ R $15$ $15$ $15$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
$31$31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
31.10.8.1$x^{10} - 20491 x^{5} + 239127552$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$