Properties

Label 15.5.388863829589238784.1
Degree $15$
Signature $[5, 5]$
Discriminant $-3.889\times 10^{17}$
Root discriminant $14.88$
Ramified primes $2, 11$
Class number $1$
Class group trivial
Galois group $S_3 \times C_5$ (as 15T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 2*x^13 - x^12 + 5*x^11 - 22*x^9 + 33*x^8 - 22*x^7 + 22*x^6 - 22*x^5 - 9*x^4 + 38*x^3 - 29*x^2 + 9*x - 1)
 
gp: K = bnfinit(x^15 - 3*x^14 + 2*x^13 - x^12 + 5*x^11 - 22*x^9 + 33*x^8 - 22*x^7 + 22*x^6 - 22*x^5 - 9*x^4 + 38*x^3 - 29*x^2 + 9*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 9, -29, 38, -9, -22, 22, -22, 33, -22, 0, 5, -1, 2, -3, 1]);
 

\( x^{15} - 3 x^{14} + 2 x^{13} - x^{12} + 5 x^{11} - 22 x^{9} + 33 x^{8} - 22 x^{7} + 22 x^{6} - 22 x^{5} - 9 x^{4} + 38 x^{3} - 29 x^{2} + 9 x - 1 \)

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

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-388863829589238784\)\(\medspace = -\,2^{10}\cdot 11^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $14.88$
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:  $2, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $5$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{86} a^{13} - \frac{10}{43} a^{12} + \frac{1}{43} a^{11} + \frac{7}{43} a^{10} - \frac{5}{43} a^{9} - \frac{16}{43} a^{8} + \frac{9}{86} a^{7} + \frac{5}{43} a^{6} - \frac{27}{86} a^{5} - \frac{19}{43} a^{4} - \frac{16}{43} a^{2} + \frac{23}{86} a - \frac{16}{43}$, $\frac{1}{86} a^{14} - \frac{11}{86} a^{12} + \frac{11}{86} a^{11} + \frac{6}{43} a^{10} - \frac{17}{86} a^{9} - \frac{29}{86} a^{8} - \frac{25}{86} a^{7} - \frac{21}{43} a^{6} + \frac{12}{43} a^{5} - \frac{29}{86} a^{4} + \frac{11}{86} a^{3} - \frac{15}{86} a^{2} + \frac{41}{86} a + \frac{5}{86}$

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:  $9$
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:  \( 803.578266232599 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{5}\cdot 803.578266232599 \cdot 1}{2\sqrt{388863829589238784}}\approx 0.201905770915894$

Galois group

$C_5\times S_3$ (as 15T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 30
The 15 conjugacy class representatives for $S_3 \times C_5$
Character table for $S_3 \times C_5$

Intermediate fields

3.1.484.1, \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R $15$ $15$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ $15$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.15.14.1$x^{15} - 11$$15$$1$$14$$S_3 \times C_5$$[\ ]_{15}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.44.10t1.a.a$1$ $ 2^{2} \cdot 11 $ 10.0.219503494144.1 $C_{10}$ (as 10T1) $0$ $-1$
1.44.10t1.a.b$1$ $ 2^{2} \cdot 11 $ 10.0.219503494144.1 $C_{10}$ (as 10T1) $0$ $-1$
1.44.10t1.a.c$1$ $ 2^{2} \cdot 11 $ 10.0.219503494144.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.44.10t1.a.d$1$ $ 2^{2} \cdot 11 $ 10.0.219503494144.1 $C_{10}$ (as 10T1) $0$ $-1$
* 2.484.3t2.b.a$2$ $ 2^{2} \cdot 11^{2}$ 3.1.484.1 $S_3$ (as 3T2) $1$ $0$
* 2.484.15t4.b.a$2$ $ 2^{2} \cdot 11^{2}$ 15.5.388863829589238784.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.484.15t4.b.b$2$ $ 2^{2} \cdot 11^{2}$ 15.5.388863829589238784.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.484.15t4.b.c$2$ $ 2^{2} \cdot 11^{2}$ 15.5.388863829589238784.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.484.15t4.b.d$2$ $ 2^{2} \cdot 11^{2}$ 15.5.388863829589238784.1 $S_3 \times C_5$ (as 15T4) $0$ $0$

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 *.