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Properties

Label 15.1.35656154243...5536.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 619^{7}$
Root discriminant $31.88$
Ramified primes $2, 619$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{15}$ (as 15T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 286, 669, 1003, 1473, 1337, 754, 159, -99, 40, 47, 57, -11, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 3*x^13 - 11*x^12 + 57*x^11 + 47*x^10 + 40*x^9 - 99*x^8 + 159*x^7 + 754*x^6 + 1337*x^5 + 1473*x^4 + 1003*x^3 + 669*x^2 + 286*x + 121)
 
gp: K = bnfinit(x^15 - 2*x^14 - 3*x^13 - 11*x^12 + 57*x^11 + 47*x^10 + 40*x^9 - 99*x^8 + 159*x^7 + 754*x^6 + 1337*x^5 + 1473*x^4 + 1003*x^3 + 669*x^2 + 286*x + 121, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 3 x^{13} - 11 x^{12} + 57 x^{11} + 47 x^{10} + 40 x^{9} - 99 x^{8} + 159 x^{7} + 754 x^{6} + 1337 x^{5} + 1473 x^{4} + 1003 x^{3} + 669 x^{2} + 286 x + 121 \)

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:  \(-35656154243640332225536=-\,2^{10}\cdot 619^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.88$
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:  $2, 619$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{36} a^{11} + \frac{1}{18} a^{10} - \frac{1}{36} a^{9} + \frac{1}{36} a^{8} + \frac{1}{18} a^{7} - \frac{5}{36} a^{6} - \frac{11}{36} a^{5} + \frac{7}{18} a^{4} - \frac{17}{36} a^{3} + \frac{11}{36} a^{2} - \frac{5}{18} a + \frac{13}{36}$, $\frac{1}{1188} a^{12} + \frac{1}{1188} a^{11} - \frac{1}{396} a^{10} + \frac{8}{297} a^{9} - \frac{179}{1188} a^{8} + \frac{41}{1188} a^{7} + \frac{7}{99} a^{6} + \frac{481}{1188} a^{5} + \frac{437}{1188} a^{4} + \frac{47}{594} a^{3} + \frac{7}{36} a^{2} + \frac{179}{1188} a - \frac{29}{108}$, $\frac{1}{7128} a^{13} - \frac{1}{1782} a^{11} + \frac{35}{7128} a^{10} + \frac{185}{7128} a^{9} + \frac{5}{162} a^{8} - \frac{353}{7128} a^{7} - \frac{1187}{7128} a^{6} - \frac{23}{81} a^{5} - \frac{2323}{7128} a^{4} - \frac{1447}{7128} a^{3} + \frac{185}{1782} a^{2} - \frac{83}{1188} a - \frac{43}{648}$, $\frac{1}{722629512} a^{14} + \frac{35927}{722629512} a^{13} + \frac{7969}{90328689} a^{12} + \frac{1347877}{240876504} a^{11} + \frac{319637}{10948932} a^{10} + \frac{24133283}{722629512} a^{9} + \frac{8878129}{240876504} a^{8} + \frac{90541}{10948932} a^{7} + \frac{6002243}{80292168} a^{6} + \frac{336655141}{722629512} a^{5} + \frac{2085728}{30109563} a^{4} + \frac{22726139}{80292168} a^{3} - \frac{6070229}{32846796} a^{2} + \frac{336227317}{722629512} a - \frac{13052657}{65693592}$

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:  $7$
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:  \( 1075964.47617 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{15}$ (as 15T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.2476.1, 5.1.383161.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
619Data not computed