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Properties

Label 15.1.12010486967...9375.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,5^{10}\cdot 103^{7}$
Root discriminant $25.43$
Ramified primes $5, 103$
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![-257, 1049, -1866, 2255, -2077, 1697, -1306, 867, -520, 281, -154, 81, -35, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 15*x^13 - 35*x^12 + 81*x^11 - 154*x^10 + 281*x^9 - 520*x^8 + 867*x^7 - 1306*x^6 + 1697*x^5 - 2077*x^4 + 2255*x^3 - 1866*x^2 + 1049*x - 257)
 
gp: K = bnfinit(x^15 - 4*x^14 + 15*x^13 - 35*x^12 + 81*x^11 - 154*x^10 + 281*x^9 - 520*x^8 + 867*x^7 - 1306*x^6 + 1697*x^5 - 2077*x^4 + 2255*x^3 - 1866*x^2 + 1049*x - 257, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} + 15 x^{13} - 35 x^{12} + 81 x^{11} - 154 x^{10} + 281 x^{9} - 520 x^{8} + 867 x^{7} - 1306 x^{6} + 1697 x^{5} - 2077 x^{4} + 2255 x^{3} - 1866 x^{2} + 1049 x - 257 \)

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:  \(-1201048696703974609375=-\,5^{10}\cdot 103^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.43$
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:  $5, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{33} a^{11} + \frac{4}{33} a^{10} - \frac{5}{33} a^{9} + \frac{5}{33} a^{7} + \frac{2}{33} a^{6} + \frac{7}{33} a^{5} - \frac{4}{11} a^{3} - \frac{1}{11} a^{2} + \frac{7}{33} a - \frac{1}{11}$, $\frac{1}{495} a^{12} + \frac{1}{99} a^{11} - \frac{67}{495} a^{10} - \frac{71}{495} a^{9} + \frac{71}{495} a^{8} - \frac{37}{495} a^{7} - \frac{46}{495} a^{6} - \frac{158}{495} a^{5} - \frac{67}{495} a^{4} + \frac{19}{99} a^{3} - \frac{8}{99} a^{2} + \frac{4}{495} a - \frac{14}{495}$, $\frac{1}{495} a^{13} - \frac{2}{495} a^{11} - \frac{4}{55} a^{10} - \frac{8}{165} a^{9} - \frac{62}{495} a^{8} - \frac{71}{495} a^{7} - \frac{26}{165} a^{6} - \frac{4}{15} a^{5} + \frac{20}{99} a^{4} + \frac{1}{9} a^{3} - \frac{2}{15} a^{2} + \frac{101}{495} a - \frac{40}{99}$, $\frac{1}{65040525} a^{14} + \frac{34454}{65040525} a^{13} + \frac{62122}{65040525} a^{12} - \frac{875869}{65040525} a^{11} - \frac{648019}{7226725} a^{10} + \frac{9748198}{65040525} a^{9} - \frac{7127}{289069} a^{8} + \frac{882619}{13008105} a^{7} + \frac{1150333}{7226725} a^{6} - \frac{137644}{13008105} a^{5} - \frac{2407562}{7226725} a^{4} + \frac{10332404}{65040525} a^{3} - \frac{9680113}{65040525} a^{2} - \frac{3802364}{13008105} a - \frac{28614226}{65040525}$

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:  \( 72424.3618661 \)
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

3.1.2575.1, 5.1.10609.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $15$

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
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$103$$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$