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Properties

Label 15.3.19404360556...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{22}\cdot 5^{10}\cdot 7^{10}\cdot 109^{6}$
Root discriminant $193.13$
Ramified primes $2, 5, 7, 109$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $\GL(2,4):C_2$ (as 15T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-944784, -1154736, 1125576, -69660, -230688, -202366, 13744, 137212, 1968, 3848, 152, 681, -16, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 24*x^13 - 16*x^12 + 681*x^11 + 152*x^10 + 3848*x^9 + 1968*x^8 + 137212*x^7 + 13744*x^6 - 202366*x^5 - 230688*x^4 - 69660*x^3 + 1125576*x^2 - 1154736*x - 944784)
 
gp: K = bnfinit(x^15 + 24*x^13 - 16*x^12 + 681*x^11 + 152*x^10 + 3848*x^9 + 1968*x^8 + 137212*x^7 + 13744*x^6 - 202366*x^5 - 230688*x^4 - 69660*x^3 + 1125576*x^2 - 1154736*x - 944784, 1)
 

Normalized defining polynomial

\( x^{15} + 24 x^{13} - 16 x^{12} + 681 x^{11} + 152 x^{10} + 3848 x^{9} + 1968 x^{8} + 137212 x^{7} + 13744 x^{6} - 202366 x^{5} - 230688 x^{4} - 69660 x^{3} + 1125576 x^{2} - 1154736 x - 944784 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19404360556860992487792640000000000=2^{22}\cdot 5^{10}\cdot 7^{10}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $193.13$
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:  $2, 5, 7, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{36} a^{11} - \frac{1}{12} a^{9} + \frac{1}{18} a^{8} + \frac{1}{6} a^{7} + \frac{2}{9} a^{6} + \frac{7}{18} a^{5} - \frac{1}{3} a^{4} - \frac{1}{18} a^{3} - \frac{2}{9} a^{2} + \frac{2}{9} a$, $\frac{1}{324} a^{12} + \frac{2}{27} a^{10} - \frac{4}{81} a^{9} + \frac{11}{108} a^{8} + \frac{38}{81} a^{7} - \frac{10}{81} a^{6} + \frac{2}{27} a^{5} + \frac{40}{81} a^{4} + \frac{34}{81} a^{3} + \frac{67}{162} a^{2}$, $\frac{1}{2916} a^{13} + \frac{2}{243} a^{11} - \frac{4}{729} a^{10} + \frac{227}{972} a^{9} + \frac{38}{729} a^{8} + \frac{233}{729} a^{7} - \frac{79}{243} a^{6} + \frac{40}{729} a^{5} - \frac{209}{729} a^{4} - \frac{581}{1458} a^{3} - \frac{1}{9} a^{2} + \frac{1}{9} a$, $\frac{1}{48937517892153010566974881472152156344} a^{14} + \frac{56169204137108012446364574955517}{1359375497004250293527080040893115454} a^{13} + \frac{5406893370930182453501761699621123}{8156252982025501761162480245358692724} a^{12} + \frac{8461861714631407813501065327328795}{6117189736519126320871860184019019543} a^{11} - \frac{1879065477725702364201949768162656535}{16312505964051003522324960490717385448} a^{10} + \frac{1262448553394392929857070939552626711}{12234379473038252641743720368038039086} a^{9} - \frac{1067311912562286337915561691113159850}{6117189736519126320871860184019019543} a^{8} - \frac{158606355010574164327337443413638191}{4078126491012750880581240122679346362} a^{7} - \frac{4352377663208967788769504921494401451}{12234379473038252641743720368038039086} a^{6} + \frac{706518365340704606300100853788957026}{6117189736519126320871860184019019543} a^{5} - \frac{11085011883174076417118008768846379981}{24468758946076505283487440736076078172} a^{4} - \frac{7102364145556090966958627339939140}{52283672961701934366426155418965979} a^{3} + \frac{11200164691462339216303016589125083}{50347240629787047908410371884930202} a^{2} + \frac{2038645666846581520283034445462096}{8391206771631174651401728647488367} a - \frac{123221372988713697653402882863146}{932356307959019405711303183054263}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  $8$
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:  \( 308019472964 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_5$ (as 15T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 12 conjugacy class representatives for $\GL(2,4):C_2$
Character table for $\GL(2,4):C_2$

Intermediate fields

3.3.9800.1, 5.1.380192.1

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 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ $15$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.12.8.1$x^{12} - 30 x^{9} + 175 x^{6} + 500 x^{3} + 5000$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$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$$[\ ]_{3}^{3}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.3.2.3$x^{3} - 3924$$3$$1$$2$$C_3$$[\ ]_{3}$
109.6.4.2$x^{6} - 109 x^{3} + 71286$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$