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Properties

Label 15.1.13845573412...4263.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,3^{17}\cdot 101^{7}$
Root discriminant $29.93$
Ramified primes $3, 101$
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![27, 108, 18, -264, 117, 87, -258, 216, 54, -116, 108, -24, -16, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 15*x^13 - 16*x^12 - 24*x^11 + 108*x^10 - 116*x^9 + 54*x^8 + 216*x^7 - 258*x^6 + 87*x^5 + 117*x^4 - 264*x^3 + 18*x^2 + 108*x + 27)
 
gp: K = bnfinit(x^15 - 6*x^14 + 15*x^13 - 16*x^12 - 24*x^11 + 108*x^10 - 116*x^9 + 54*x^8 + 216*x^7 - 258*x^6 + 87*x^5 + 117*x^4 - 264*x^3 + 18*x^2 + 108*x + 27, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} + 15 x^{13} - 16 x^{12} - 24 x^{11} + 108 x^{10} - 116 x^{9} + 54 x^{8} + 216 x^{7} - 258 x^{6} + 87 x^{5} + 117 x^{4} - 264 x^{3} + 18 x^{2} + 108 x + 27 \)

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:  \(-13845573412916166484263=-\,3^{17}\cdot 101^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.93$
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:  $3, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{105} a^{12} + \frac{12}{35} a^{11} - \frac{34}{105} a^{9} - \frac{16}{35} a^{8} + \frac{2}{7} a^{7} + \frac{34}{105} a^{6} - \frac{9}{35} a^{4} + \frac{13}{35} a^{3} + \frac{3}{35} a^{2} - \frac{17}{35} a - \frac{11}{35}$, $\frac{1}{2415} a^{13} - \frac{4}{2415} a^{12} - \frac{68}{161} a^{11} + \frac{281}{2415} a^{10} - \frac{998}{2415} a^{9} - \frac{45}{161} a^{8} - \frac{641}{2415} a^{7} - \frac{125}{483} a^{6} + \frac{131}{805} a^{5} + \frac{58}{805} a^{4} + \frac{43}{805} a^{3} - \frac{32}{805} a^{2} + \frac{284}{805} a - \frac{10}{161}$, $\frac{1}{5706357615} a^{14} - \frac{29893}{1902119205} a^{13} - \frac{162438}{634039735} a^{12} + \frac{828732539}{5706357615} a^{11} - \frac{169651777}{634039735} a^{10} - \frac{41731339}{1902119205} a^{9} - \frac{65584843}{1141271523} a^{8} + \frac{7273928}{380423841} a^{7} - \frac{36458593}{380423841} a^{6} + \frac{107641378}{1902119205} a^{5} + \frac{838368536}{1902119205} a^{4} - \frac{304078106}{634039735} a^{3} + \frac{195121433}{1902119205} a^{2} + \frac{32747164}{90577105} a - \frac{259314576}{634039735}$

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:  \( 359715.566347 \)
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

3.1.2727.1, 5.1.91809.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ $15$

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
$3$3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$