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Properties

Label 15.1.63623662184...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 3^{13}\cdot 5^{9}\cdot 37^{13}$
Root discriminant $450.38$
Ramified primes $2, 3, 5, 37$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1005343698, 504130986, 136789686, -30340590, -17515380, -3669519, -248742, -526083, -115242, 49272, 6410, -304, 812, 97, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 97*x^13 + 812*x^12 - 304*x^11 + 6410*x^10 + 49272*x^9 - 115242*x^8 - 526083*x^7 - 248742*x^6 - 3669519*x^5 - 17515380*x^4 - 30340590*x^3 + 136789686*x^2 + 504130986*x + 1005343698)
 
gp: K = bnfinit(x^15 - 2*x^14 + 97*x^13 + 812*x^12 - 304*x^11 + 6410*x^10 + 49272*x^9 - 115242*x^8 - 526083*x^7 - 248742*x^6 - 3669519*x^5 - 17515380*x^4 - 30340590*x^3 + 136789686*x^2 + 504130986*x + 1005343698, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} + 97 x^{13} + 812 x^{12} - 304 x^{11} + 6410 x^{10} + 49272 x^{9} - 115242 x^{8} - 526083 x^{7} - 248742 x^{6} - 3669519 x^{5} - 17515380 x^{4} - 30340590 x^{3} + 136789686 x^{2} + 504130986 x + 1005343698 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6362366218401470792736132808704000000000=-\,2^{23}\cdot 3^{13}\cdot 5^{9}\cdot 37^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $450.38$
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:  $2, 3, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{10} a^{11} + \frac{1}{5} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{7} - \frac{1}{10} a^{5} + \frac{2}{5} a^{3} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{10} a^{12} - \frac{3}{10} a^{9} - \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{100} a^{13} - \frac{1}{25} a^{11} + \frac{1}{25} a^{10} - \frac{3}{25} a^{9} - \frac{2}{25} a^{8} + \frac{19}{50} a^{7} + \frac{21}{50} a^{6} - \frac{17}{100} a^{5} + \frac{8}{25} a^{4} + \frac{8}{25} a^{3} - \frac{12}{25} a^{2} + \frac{1}{50} a + \frac{19}{50}$, $\frac{1}{1061788634994238471810633773229611793663825219127475100} a^{14} - \frac{2259319652642578239824149552549964216120622730376183}{530894317497119235905316886614805896831912609563737550} a^{13} - \frac{5343256046321031116153565111042797931996943988652071}{265447158748559617952658443307402948415956304781868775} a^{12} + \frac{13588295824015336992682280307428366370350800175798039}{530894317497119235905316886614805896831912609563737550} a^{11} - \frac{107939275242647936020727055140214500368214739320882253}{530894317497119235905316886614805896831912609563737550} a^{10} + \frac{17695999675270371313984966242997480747432259450774391}{265447158748559617952658443307402948415956304781868775} a^{9} - \frac{69443369734876804148226238660050857790868907189921677}{530894317497119235905316886614805896831912609563737550} a^{8} - \frac{103603368214291354147253892673278433457670988258590984}{265447158748559617952658443307402948415956304781868775} a^{7} + \frac{365213670497502443949365468538930674345101550587832041}{1061788634994238471810633773229611793663825219127475100} a^{6} + \frac{249962379580561565960536963934245539981792265467781567}{530894317497119235905316886614805896831912609563737550} a^{5} + \frac{214796823193950249170515346774430518980389493423392}{10617886349942384718106337732296117936638252191274751} a^{4} - \frac{15536940211092571239741751667466133876713865831365829}{53089431749711923590531688661480589683191260956373755} a^{3} + \frac{5057430777784328908983298845805520499725688296987663}{106178863499423847181063377322961179366382521912747510} a^{2} - \frac{255627425620549280524962300765209582758199225176268637}{530894317497119235905316886614805896831912609563737550} a - \frac{423149727156509976039223011501495408695026778639037}{2628189690579798197551073696112900479365904007741275}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  $7$
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:  \( 33354685093488.18 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.888.1, 5.1.303614082000.6

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

Sibling fields

Degree 30 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 R R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$37$37.5.4.1$x^{5} - 37$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
37.10.9.2$x^{10} + 74$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$