Properties

Label 15.1.494...375.1
Degree $15$
Signature $[1, 7]$
Discriminant $-4.947\times 10^{18}$
Root discriminant $17.63$
Ramified primes $5, 47$
Class number $1$
Class group trivial
Galois group $D_{15}$ (as 15T2)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^12 + 9*x^11 + 20*x^10 + 57*x^9 + 29*x^8 + 27*x^7 - 53*x^6 - 44*x^5 - 84*x^4 - 43*x^3 - 100*x^2 - 75*x - 55)
 
gp: K = bnfinit(x^15 - 6*x^12 + 9*x^11 + 20*x^10 + 57*x^9 + 29*x^8 + 27*x^7 - 53*x^6 - 44*x^5 - 84*x^4 - 43*x^3 - 100*x^2 - 75*x - 55, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-55, -75, -100, -43, -84, -44, -53, 27, 29, 57, 20, 9, -6, 0, 0, 1]);
 

\(x^{15} - 6 x^{12} + 9 x^{11} + 20 x^{10} + 57 x^{9} + 29 x^{8} + 27 x^{7} - 53 x^{6} - 44 x^{5} - 84 x^{4} - 43 x^{3} - 100 x^{2} - 75 x - 55\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-4947491410771484375\)\(\medspace = -\,5^{10}\cdot 47^{7}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.63$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 47$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{235} a^{12} + \frac{31}{235} a^{11} - \frac{26}{235} a^{10} + \frac{89}{235} a^{9} + \frac{8}{235} a^{8} + \frac{107}{235} a^{7} - \frac{10}{47} a^{6} + \frac{2}{5} a^{5} + \frac{37}{235} a^{4} + \frac{36}{235} a^{3} + \frac{17}{47} a^{2} - \frac{4}{47} a + \frac{19}{47}$, $\frac{1}{235} a^{13} - \frac{1}{5} a^{11} - \frac{9}{47} a^{10} + \frac{69}{235} a^{9} + \frac{2}{5} a^{8} - \frac{77}{235} a^{7} - \frac{1}{235} a^{6} - \frac{57}{235} a^{5} + \frac{64}{235} a^{4} - \frac{91}{235} a^{3} - \frac{14}{47} a^{2} + \frac{2}{47} a + \frac{22}{47}$, $\frac{1}{468523025} a^{14} + \frac{241037}{468523025} a^{13} - \frac{13446}{468523025} a^{12} + \frac{112445122}{468523025} a^{11} - \frac{115721117}{468523025} a^{10} + \frac{24747486}{468523025} a^{9} + \frac{161566574}{468523025} a^{8} - \frac{177333328}{468523025} a^{7} + \frac{12454981}{468523025} a^{6} - \frac{122483791}{468523025} a^{5} - \frac{64219836}{468523025} a^{4} - \frac{33727116}{468523025} a^{3} - \frac{22786853}{93704605} a^{2} - \frac{9170231}{93704605} a + \frac{15014348}{93704605}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1727.98179688 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{7}\cdot 1727.98179688 \cdot 1}{2\sqrt{4947491410771484375}}\approx 0.300334841565$

Galois group

$D_{15}$ (as 15T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.1175.1, 5.1.2209.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: Deg 30

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15$ $15$ R $15$ ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ $15$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ R ${\href{/padicField/53.5.0.1}{5} }^{3}$ $15$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

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$$[\ ]_{3}^{2}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.47.2t1.a.a$1$ $ 47 $ \(\Q(\sqrt{-47}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1175.3t2.a.a$2$ $ 5^{2} \cdot 47 $ 3.1.1175.1 $S_3$ (as 3T2) $1$ $0$
* 2.47.5t2.a.b$2$ $ 47 $ 5.1.2209.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.47.5t2.a.a$2$ $ 47 $ 5.1.2209.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.1175.15t2.a.d$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.1175.15t2.a.b$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.1175.15t2.a.c$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.1175.15t2.a.a$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.