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Properties

Label 15.15.1386229293...6697.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{15}\cdot 11^{13}\cdot 23^{4}$
Root discriminant $55.31$
Ramified primes $3, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T33

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-197317, -991875, -1909161, -1591761, -170775, 652257, 386911, -19527, -79524, -15374, 5532, 1854, -123, -75, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 75*x^13 - 123*x^12 + 1854*x^11 + 5532*x^10 - 15374*x^9 - 79524*x^8 - 19527*x^7 + 386911*x^6 + 652257*x^5 - 170775*x^4 - 1591761*x^3 - 1909161*x^2 - 991875*x - 197317)
 
gp: K = bnfinit(x^15 - 75*x^13 - 123*x^12 + 1854*x^11 + 5532*x^10 - 15374*x^9 - 79524*x^8 - 19527*x^7 + 386911*x^6 + 652257*x^5 - 170775*x^4 - 1591761*x^3 - 1909161*x^2 - 991875*x - 197317, 1)
 

Normalized defining polynomial

\( x^{15} - 75 x^{13} - 123 x^{12} + 1854 x^{11} + 5532 x^{10} - 15374 x^{9} - 79524 x^{8} - 19527 x^{7} + 386911 x^{6} + 652257 x^{5} - 170775 x^{4} - 1591761 x^{3} - 1909161 x^{2} - 991875 x - 197317 \)

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:  \(138622929316925604547946697=3^{15}\cdot 11^{13}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.31$
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:  $3, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} - \frac{6}{23} a^{10} - \frac{8}{23} a^{9} - \frac{9}{23} a^{8} - \frac{11}{23} a^{7} - \frac{10}{23} a^{6} + \frac{10}{23} a^{5} + \frac{5}{23} a^{3}$, $\frac{1}{23} a^{13} - \frac{6}{23} a^{11} - \frac{8}{23} a^{10} - \frac{9}{23} a^{9} - \frac{11}{23} a^{8} - \frac{10}{23} a^{7} + \frac{10}{23} a^{6} + \frac{5}{23} a^{4}$, $\frac{1}{142246152185314722620989} a^{14} + \frac{1798646772246468991179}{142246152185314722620989} a^{13} - \frac{2516559212023943632978}{142246152185314722620989} a^{12} - \frac{65861314031020933039413}{142246152185314722620989} a^{11} + \frac{60722298689155043408716}{142246152185314722620989} a^{10} + \frac{55051161193458383546271}{142246152185314722620989} a^{9} + \frac{7896788332046603868345}{142246152185314722620989} a^{8} + \frac{54413102618664943132387}{142246152185314722620989} a^{7} - \frac{44109143253565810696658}{142246152185314722620989} a^{6} - \frac{55462432988439340926472}{142246152185314722620989} a^{5} + \frac{58955209326650974363149}{142246152185314722620989} a^{4} + \frac{61673585276561651532457}{142246152185314722620989} a^{3} + \frac{269133178435997183520}{6184615312404987940043} a^{2} - \frac{1626118907849408668083}{6184615312404987940043} a + \frac{2381774337123534083834}{6184615312404987940043}$

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:  \( 164409576.613 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T33:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 810
The 18 conjugacy class representatives for [3^4:2]5
Character table for [3^4:2]5

Intermediate fields

\(\Q(\zeta_{11})^+\)

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

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 siblings: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
3Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$