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Properties

Label 15.1.37362650893...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 5^{28}\cdot 11^{12}\cdot 199^{5}$
Root discriminant $1273.00$
Ramified primes $2, 5, 11, 199$
Class number $150$ (GRH)
Class group $[5, 30]$ (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![-9329068, -19039100, -4003430, 4878805, -3565970, 186450, 309215, -139405, -29310, -1125, 794, 565, -25, -15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 15*x^13 - 25*x^12 + 565*x^11 + 794*x^10 - 1125*x^9 - 29310*x^8 - 139405*x^7 + 309215*x^6 + 186450*x^5 - 3565970*x^4 + 4878805*x^3 - 4003430*x^2 - 19039100*x - 9329068)
 
gp: K = bnfinit(x^15 - 5*x^14 - 15*x^13 - 25*x^12 + 565*x^11 + 794*x^10 - 1125*x^9 - 29310*x^8 - 139405*x^7 + 309215*x^6 + 186450*x^5 - 3565970*x^4 + 4878805*x^3 - 4003430*x^2 - 19039100*x - 9329068, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 15 x^{13} - 25 x^{12} + 565 x^{11} + 794 x^{10} - 1125 x^{9} - 29310 x^{8} - 139405 x^{7} + 309215 x^{6} + 186450 x^{5} - 3565970 x^{4} + 4878805 x^{3} - 4003430 x^{2} - 19039100 x - 9329068 \)

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:  \(-37362650893059880826731834411621093750000000000=-\,2^{10}\cdot 5^{28}\cdot 11^{12}\cdot 199^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1273.00$
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:  $2, 5, 11, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{50} a^{10} - \frac{1}{5} a^{9} - \frac{1}{2} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{25} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a - \frac{4}{25}$, $\frac{1}{50} a^{11} - \frac{1}{2} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{25} a^{6} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{5} a^{2} - \frac{4}{25} a + \frac{2}{5}$, $\frac{1}{50} a^{12} - \frac{1}{5} a^{9} - \frac{3}{10} a^{8} - \frac{1}{25} a^{7} + \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{5} a^{3} + \frac{17}{50} a^{2} + \frac{2}{5} a$, $\frac{1}{50} a^{13} - \frac{3}{10} a^{9} - \frac{1}{25} a^{8} + \frac{1}{5} a^{6} + \frac{1}{10} a^{5} - \frac{1}{5} a^{4} + \frac{17}{50} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{109172216066820949342629470089518469750} a^{14} - \frac{326950552404206509285616900125922038}{54586108033410474671314735044759234875} a^{13} - \frac{4618958843428505038052122037676467}{54586108033410474671314735044759234875} a^{12} - \frac{158633061879951798717470597964186278}{54586108033410474671314735044759234875} a^{11} - \frac{51884796177463037295872398763092127}{54586108033410474671314735044759234875} a^{10} + \frac{19944518285608887765347826278682706584}{54586108033410474671314735044759234875} a^{9} + \frac{21757500033575109725550752943543421847}{109172216066820949342629470089518469750} a^{8} + \frac{19357860383352485880681193716172603529}{54586108033410474671314735044759234875} a^{7} - \frac{15592771926391248949465650441349649183}{109172216066820949342629470089518469750} a^{6} + \frac{15428148364639962145854904145857204524}{54586108033410474671314735044759234875} a^{5} - \frac{6772419167588157852264970728678819413}{109172216066820949342629470089518469750} a^{4} - \frac{20899654279430970735306831015283058511}{54586108033410474671314735044759234875} a^{3} - \frac{12290862627021675957194846695782160763}{109172216066820949342629470089518469750} a^{2} - \frac{24952631780189691057737960492482105236}{54586108033410474671314735044759234875} a - \frac{16726972846256882824668884069056099139}{54586108033410474671314735044759234875}$

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

Class group and class number

$C_{5}\times C_{30}$, which has order $150$ (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:  \( 3549486217575014.5 \) (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

3.1.3980.1, 5.1.28595703125.5

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.5$x^{10} + 55$$10$$1$$19$$F_5$$[9/4]_{4}$
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.4$x^{10} - 781 x^{5} + 290521$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$