Properties

Label 15.1.677952124826430464.1
Degree $15$
Signature $[1, 7]$
Discriminant $-6.780\times 10^{17}$
Root discriminant $15.44$
Ramified primes $2, 131$
Class number $1$
Class group trivial
Galois group $D_{15}$ (as 15T2)

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Normalized defining polynomial

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)
 
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]);
 

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

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:  \(-677952124826430464\)\(\medspace = -\,2^{10}\cdot 131^{7}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.44$
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:  $2, 131$
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 \(a\))

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

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 $2$)
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:  \( 1343.20606196 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{7}\cdot 1343.20606196 \cdot 1}{2\sqrt{677952124826430464}}\approx 0.630670051922$

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

3.1.524.1, 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: 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 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.

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
$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}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.131.2t1.a.a$1$ $ 131 $ \(\Q(\sqrt{-131}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.524.3t2.a.a$2$ $ 2^{2} \cdot 131 $ 3.1.524.1 $S_3$ (as 3T2) $1$ $0$
* 2.131.5t2.a.a$2$ $ 131 $ 5.1.17161.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.131.5t2.a.b$2$ $ 131 $ 5.1.17161.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.524.15t2.a.c$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.524.15t2.a.a$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.524.15t2.a.b$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.524.15t2.a.d$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (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 *.