↑

Properties

Label 15.1.19147985200...8192.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 787^{7}$
Root discriminant $35.66$
Ramified primes $2, 787$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{15}$ (as 15T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -272, 656, 4896, 6224, 2920, 1864, 168, 600, 276, 80, -2, 17, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 9*x^13 + 17*x^12 - 2*x^11 + 80*x^10 + 276*x^9 + 600*x^8 + 168*x^7 + 1864*x^6 + 2920*x^5 + 6224*x^4 + 4896*x^3 + 656*x^2 - 272*x + 16)
 
gp: K = bnfinit(x^15 - 3*x^14 + 9*x^13 + 17*x^12 - 2*x^11 + 80*x^10 + 276*x^9 + 600*x^8 + 168*x^7 + 1864*x^6 + 2920*x^5 + 6224*x^4 + 4896*x^3 + 656*x^2 - 272*x + 16, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} + 9 x^{13} + 17 x^{12} - 2 x^{11} + 80 x^{10} + 276 x^{9} + 600 x^{8} + 168 x^{7} + 1864 x^{6} + 2920 x^{5} + 6224 x^{4} + 4896 x^{3} + 656 x^{2} - 272 x + 16 \)

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:  \(-191479852009379288968192=-\,2^{10}\cdot 787^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.66$
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:  $2, 787$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{5}{12} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{120} a^{9} - \frac{1}{24} a^{8} - \frac{1}{40} a^{7} - \frac{9}{40} a^{6} + \frac{7}{20} a^{5} - \frac{7}{20} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{7}{15} a - \frac{13}{30}$, $\frac{1}{360} a^{10} + \frac{1}{360} a^{9} - \frac{13}{360} a^{8} - \frac{1}{72} a^{7} - \frac{1}{18} a^{6} + \frac{7}{36} a^{5} - \frac{7}{18} a^{4} - \frac{7}{18} a^{3} + \frac{1}{15} a^{2} - \frac{13}{30} a + \frac{1}{45}$, $\frac{1}{360} a^{11} + \frac{1}{360} a^{9} - \frac{7}{360} a^{8} - \frac{5}{24} a^{6} - \frac{1}{6} a^{5} + \frac{5}{12} a^{4} - \frac{17}{45} a^{3} + \frac{1}{6} a^{2} + \frac{41}{90} a - \frac{17}{90}$, $\frac{1}{720} a^{12} - \frac{1}{720} a^{11} - \frac{1}{720} a^{10} - \frac{1}{720} a^{9} - \frac{1}{60} a^{8} + \frac{7}{180} a^{7} - \frac{17}{180} a^{6} + \frac{13}{45} a^{5} - \frac{11}{30} a^{4} + \frac{41}{180} a^{3} + \frac{31}{90} a^{2} + \frac{43}{90} a + \frac{4}{45}$, $\frac{1}{168480} a^{13} - \frac{113}{168480} a^{12} - \frac{7}{18720} a^{11} - \frac{1}{33696} a^{10} + \frac{37}{21060} a^{9} - \frac{7}{1080} a^{8} + \frac{2459}{84240} a^{7} + \frac{3959}{21060} a^{6} + \frac{79}{1560} a^{5} - \frac{41}{104} a^{4} + \frac{98}{5265} a^{3} - \frac{758}{1755} a^{2} + \frac{5029}{21060} a - \frac{4391}{10530}$, $\frac{1}{19712160} a^{14} - \frac{19}{19712160} a^{13} + \frac{313}{19712160} a^{12} + \frac{22387}{19712160} a^{11} - \frac{19}{164268} a^{10} + \frac{31969}{9856080} a^{9} - \frac{141763}{9856080} a^{8} + \frac{101701}{2464020} a^{7} + \frac{318439}{4928040} a^{6} + \frac{86447}{547560} a^{5} + \frac{593303}{2464020} a^{4} + \frac{268187}{2464020} a^{3} - \frac{854441}{2464020} a^{2} + \frac{1346}{3159} a + \frac{215174}{616005}$

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:  \( 3272368.23921 \) (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

3.1.3148.1, 5.1.619369.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$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}$
787Data not computed