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Properties

Label 15.1.34540821501...1343.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,7^{10}\cdot 11^{7}\cdot 13^{7}$
Root discriminant $37.09$
Ramified primes $7, 11, 13$
Class number $3$ (GRH)
Class group $[3]$ (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![-19375, 6050, -13194, -965, 2169, -6007, 615, -1462, -800, 364, -206, 86, -38, 22, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 22*x^13 - 38*x^12 + 86*x^11 - 206*x^10 + 364*x^9 - 800*x^8 - 1462*x^7 + 615*x^6 - 6007*x^5 + 2169*x^4 - 965*x^3 - 13194*x^2 + 6050*x - 19375)
 
gp: K = bnfinit(x^15 - 7*x^14 + 22*x^13 - 38*x^12 + 86*x^11 - 206*x^10 + 364*x^9 - 800*x^8 - 1462*x^7 + 615*x^6 - 6007*x^5 + 2169*x^4 - 965*x^3 - 13194*x^2 + 6050*x - 19375, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} + 22 x^{13} - 38 x^{12} + 86 x^{11} - 206 x^{10} + 364 x^{9} - 800 x^{8} - 1462 x^{7} + 615 x^{6} - 6007 x^{5} + 2169 x^{4} - 965 x^{3} - 13194 x^{2} + 6050 x - 19375 \)

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:  \(-345408215017012205401343=-\,7^{10}\cdot 11^{7}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.09$
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:  $7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a$, $\frac{1}{55} a^{11} + \frac{4}{55} a^{10} - \frac{4}{55} a^{9} - \frac{21}{55} a^{8} + \frac{2}{11} a^{7} - \frac{6}{55} a^{6} + \frac{2}{11} a^{5} + \frac{1}{55} a^{4} - \frac{2}{55} a^{3} + \frac{19}{55} a^{2} + \frac{2}{5} a + \frac{2}{11}$, $\frac{1}{55} a^{12} + \frac{2}{55} a^{10} - \frac{1}{11} a^{9} + \frac{17}{55} a^{8} - \frac{24}{55} a^{7} + \frac{1}{55} a^{6} + \frac{27}{55} a^{5} + \frac{27}{55} a^{4} + \frac{27}{55} a^{3} + \frac{1}{55} a^{2} - \frac{1}{55} a + \frac{3}{11}$, $\frac{1}{55} a^{13} - \frac{2}{55} a^{10} + \frac{3}{55} a^{9} - \frac{3}{11} a^{8} - \frac{8}{55} a^{7} - \frac{27}{55} a^{6} - \frac{4}{55} a^{5} + \frac{14}{55} a^{4} + \frac{27}{55} a^{3} - \frac{17}{55} a^{2} + \frac{4}{55} a - \frac{4}{11}$, $\frac{1}{63486873845515044534778025} a^{14} + \frac{22314214597442891366998}{63486873845515044534778025} a^{13} + \frac{492985965836091714815157}{63486873845515044534778025} a^{12} + \frac{49348861651856051002127}{63486873845515044534778025} a^{11} + \frac{2622342283597218999654031}{63486873845515044534778025} a^{10} - \frac{155099712624913135192496}{63486873845515044534778025} a^{9} - \frac{12945772109613267780297776}{63486873845515044534778025} a^{8} - \frac{3185726007132139752131976}{12697374769103008906955605} a^{7} - \frac{8395081930636697225288022}{63486873845515044534778025} a^{6} - \frac{5941398568258988965243}{19385304990996960163291} a^{5} + \frac{14755837682999405837224373}{63486873845515044534778025} a^{4} - \frac{27399069260842808156770161}{63486873845515044534778025} a^{3} + \frac{4652011471182629198958828}{12697374769103008906955605} a^{2} + \frac{4938808752992268219186416}{63486873845515044534778025} a - \frac{667329136922010890779685}{2539474953820601781391121}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 478768.310095 \) (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

3.1.7007.1, 5.1.20449.1

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 $15$ $15$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
7Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$