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Properties

Label 15.1.62030642594...5712.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 347^{7}$
Root discriminant $24.33$
Ramified primes $2, 347$
Class number $1$
Class group Trivial
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![-14, 68, -146, 163, -11, -187, 134, 269, -589, 454, -153, 3, 18, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 3*x^13 + 18*x^12 + 3*x^11 - 153*x^10 + 454*x^9 - 589*x^8 + 269*x^7 + 134*x^6 - 187*x^5 - 11*x^4 + 163*x^3 - 146*x^2 + 68*x - 14)
 
gp: K = bnfinit(x^15 - 3*x^14 - 3*x^13 + 18*x^12 + 3*x^11 - 153*x^10 + 454*x^9 - 589*x^8 + 269*x^7 + 134*x^6 - 187*x^5 - 11*x^4 + 163*x^3 - 146*x^2 + 68*x - 14, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 3 x^{13} + 18 x^{12} + 3 x^{11} - 153 x^{10} + 454 x^{9} - 589 x^{8} + 269 x^{7} + 134 x^{6} - 187 x^{5} - 11 x^{4} + 163 x^{3} - 146 x^{2} + 68 x - 14 \)

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:  \(-620306425941546355712=-\,2^{10}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.33$
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:  $2, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{35} a^{11} + \frac{16}{35} a^{10} + \frac{11}{35} a^{9} + \frac{17}{35} a^{8} + \frac{9}{35} a^{7} + \frac{1}{35} a^{6} + \frac{1}{35} a^{5} - \frac{2}{5} a^{4} - \frac{1}{7} a^{3} - \frac{13}{35} a^{2} - \frac{1}{35} a - \frac{1}{5}$, $\frac{1}{140} a^{12} - \frac{1}{70} a^{11} - \frac{8}{35} a^{10} + \frac{29}{140} a^{9} + \frac{9}{70} a^{8} + \frac{1}{10} a^{7} - \frac{17}{140} a^{6} + \frac{19}{70} a^{5} - \frac{17}{35} a^{4} + \frac{1}{20} a^{3} + \frac{29}{70} a^{2} + \frac{23}{70} a - \frac{1}{10}$, $\frac{1}{140} a^{13} - \frac{19}{140} a^{10} + \frac{13}{35} a^{9} - \frac{19}{70} a^{8} + \frac{11}{28} a^{7} + \frac{2}{7} a^{6} + \frac{11}{35} a^{5} + \frac{67}{140} a^{4} + \frac{8}{35} a^{3} - \frac{13}{70} a^{2} + \frac{3}{10} a$, $\frac{1}{1522787000} a^{14} + \frac{62533}{152278700} a^{13} - \frac{1339663}{1522787000} a^{12} + \frac{14908239}{1522787000} a^{11} - \frac{14924461}{152278700} a^{10} - \frac{43369033}{1522787000} a^{9} - \frac{102556067}{304557400} a^{8} + \frac{85164589}{380696750} a^{7} - \frac{165525233}{1522787000} a^{6} - \frac{6656931}{304557400} a^{5} + \frac{230119649}{761393500} a^{4} - \frac{17202827}{1522787000} a^{3} - \frac{110257957}{380696750} a^{2} - \frac{9920671}{21754100} a + \frac{29637797}{108770500}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 74054.3786112 \)
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

3.1.1388.1, 5.1.120409.1

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 $15$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$2.3.2.1$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}$
347Data not computed