Properties

Label 15.1.677952124826430464.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 131^{7}$
Root discriminant $15.44$
Ramified primes $2, 131$
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![4, 4, -36, 32, 28, -6, 19, -25, -12, -6, -2, 4, 0, 4, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 4*x^13 + 4*x^11 - 2*x^10 - 6*x^9 - 12*x^8 - 25*x^7 + 19*x^6 - 6*x^5 + 28*x^4 + 32*x^3 - 36*x^2 + 4*x + 4)
gp: K = bnfinit(x^15 - x^14 + 4*x^13 + 4*x^11 - 2*x^10 - 6*x^9 - 12*x^8 - 25*x^7 + 19*x^6 - 6*x^5 + 28*x^4 + 32*x^3 - 36*x^2 + 4*x + 4, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut x^{14} \) \(\mathstrut +\mathstrut 4 x^{13} \) \(\mathstrut +\mathstrut 4 x^{11} \) \(\mathstrut -\mathstrut 2 x^{10} \) \(\mathstrut -\mathstrut 6 x^{9} \) \(\mathstrut -\mathstrut 12 x^{8} \) \(\mathstrut -\mathstrut 25 x^{7} \) \(\mathstrut +\mathstrut 19 x^{6} \) \(\mathstrut -\mathstrut 6 x^{5} \) \(\mathstrut +\mathstrut 28 x^{4} \) \(\mathstrut +\mathstrut 32 x^{3} \) \(\mathstrut -\mathstrut 36 x^{2} \) \(\mathstrut +\mathstrut 4 x \) \(\mathstrut +\mathstrut 4 \)

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:  \(-677952124826430464=-\,2^{10}\cdot 131^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $15.44$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 131$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{232} a^{13} - \frac{9}{232} a^{12} + \frac{19}{232} a^{11} + \frac{13}{232} a^{10} - \frac{23}{232} a^{9} - \frac{37}{232} a^{8} - \frac{23}{232} a^{7} + \frac{19}{232} a^{6} + \frac{45}{116} a^{5} + \frac{7}{116} a^{4} - \frac{7}{58} a^{3} - \frac{3}{29} a^{2} + \frac{10}{29} a - \frac{1}{58}$, $\frac{1}{90712} a^{14} + \frac{35}{22678} a^{13} + \frac{3225}{45356} a^{12} + \frac{4293}{45356} a^{11} + \frac{24}{391} a^{10} - \frac{21}{45356} a^{9} - \frac{8829}{45356} a^{8} + \frac{10215}{45356} a^{7} + \frac{11447}{90712} a^{6} - \frac{10949}{45356} a^{5} + \frac{3092}{11339} a^{4} + \frac{4507}{11339} a^{3} - \frac{7979}{22678} a^{2} - \frac{1048}{11339} a + \frac{5535}{22678}$

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

Class group and class number

Trivial Abelian group, 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:  \( \frac{16}{493} a^{14} - \frac{33}{1972} a^{13} + \frac{61}{493} a^{12} + \frac{13}{1972} a^{11} + \frac{169}{986} a^{10} - \frac{111}{493} a^{9} - \frac{683}{1972} a^{8} - \frac{2109}{1972} a^{7} - \frac{686}{493} a^{6} - \frac{723}{1972} a^{5} + \frac{387}{1972} a^{4} + \frac{2331}{986} a^{3} + \frac{2105}{986} a^{2} + \frac{1553}{986} a - \frac{108}{493} \),  \( \frac{111}{45356} a^{14} - \frac{1373}{90712} a^{13} + \frac{449}{90712} a^{12} - \frac{109}{3128} a^{11} - \frac{2201}{90712} a^{10} - \frac{11279}{90712} a^{9} + \frac{2045}{90712} a^{8} - \frac{2095}{90712} a^{7} + \frac{19675}{90712} a^{6} - \frac{893}{22678} a^{5} + \frac{1267}{45356} a^{4} - \frac{16843}{22678} a^{3} - \frac{499}{22678} a^{2} - \frac{1575}{11339} a + \frac{13933}{22678} \),  \( \frac{1161}{90712} a^{14} - \frac{210}{11339} a^{13} + \frac{2155}{22678} a^{12} - \frac{295}{45356} a^{11} + \frac{2596}{11339} a^{10} + \frac{157}{1564} a^{9} + \frac{21883}{45356} a^{8} + \frac{5277}{45356} a^{7} + \frac{10043}{90712} a^{6} - \frac{8981}{45356} a^{5} - \frac{45583}{45356} a^{4} - \frac{9510}{11339} a^{3} - \frac{23507}{22678} a^{2} + \frac{8339}{22678} a - \frac{8943}{22678} \),  \( \frac{5303}{45356} a^{14} - \frac{2075}{11339} a^{13} + \frac{6760}{11339} a^{12} - \frac{16423}{45356} a^{11} + \frac{9471}{11339} a^{10} - \frac{8175}{11339} a^{9} + \frac{4517}{45356} a^{8} - \frac{63287}{45356} a^{7} - \frac{72817}{45356} a^{6} + \frac{68987}{22678} a^{5} - \frac{129439}{45356} a^{4} + \frac{104253}{22678} a^{3} - \frac{4091}{11339} a^{2} - \frac{88237}{22678} a + \frac{20458}{11339} \),  \( \frac{8603}{90712} a^{14} - \frac{65}{1564} a^{13} + \frac{14955}{45356} a^{12} + \frac{5079}{22678} a^{11} + \frac{18815}{45356} a^{10} + \frac{4671}{45356} a^{9} - \frac{27597}{45356} a^{8} - \frac{25195}{22678} a^{7} - \frac{9205}{3128} a^{6} + \frac{17399}{22678} a^{5} - \frac{427}{11339} a^{4} + \frac{27576}{11339} a^{3} + \frac{48600}{11339} a^{2} - \frac{18643}{11339} a - \frac{22617}{22678} \),  \( \frac{3019}{45356} a^{14} - \frac{687}{11339} a^{13} + \frac{2729}{11339} a^{12} + \frac{2135}{45356} a^{11} + \frac{4261}{22678} a^{10} - \frac{1621}{22678} a^{9} - \frac{25977}{45356} a^{8} - \frac{1029}{1564} a^{7} - \frac{80197}{45356} a^{6} + \frac{29043}{22678} a^{5} - \frac{3407}{45356} a^{4} + \frac{18043}{11339} a^{3} + \frac{30625}{11339} a^{2} - \frac{43161}{22678} a + \frac{2344}{11339} \),  \( \frac{1346}{11339} a^{14} - \frac{3607}{45356} a^{13} + \frac{19685}{45356} a^{12} + \frac{4239}{22678} a^{11} + \frac{16955}{45356} a^{10} + \frac{3389}{45356} a^{9} - \frac{22971}{22678} a^{8} - \frac{17454}{11339} a^{7} - \frac{179023}{45356} a^{6} + \frac{39581}{22678} a^{5} - \frac{10443}{45356} a^{4} + \frac{40729}{11339} a^{3} + \frac{2408}{391} a^{2} - \frac{81363}{22678} a - \frac{12137}{11339} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1343.20606196 \)
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

3.1.524.1, 5.1.17161.1

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 $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ $15$

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
$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$$[\ ]_{3}^{2}$
$131$$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$