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Properties

Label 15.7.17318268044...7136.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{10}\cdot 3^{15}\cdot 4903^{3}$
Root discriminant $26.06$
Ramified primes $2, 3, 4903$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T93

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, -144, -72, 616, -474, -297, 408, -72, 54, -88, 12, 27, -6, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^13 - 6*x^12 + 27*x^11 + 12*x^10 - 88*x^9 + 54*x^8 - 72*x^7 + 408*x^6 - 297*x^5 - 474*x^4 + 616*x^3 - 72*x^2 - 144*x + 32)
 
gp: K = bnfinit(x^15 - 6*x^13 - 6*x^12 + 27*x^11 + 12*x^10 - 88*x^9 + 54*x^8 - 72*x^7 + 408*x^6 - 297*x^5 - 474*x^4 + 616*x^3 - 72*x^2 - 144*x + 32, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{13} - 6 x^{12} + 27 x^{11} + 12 x^{10} - 88 x^{9} + 54 x^{8} - 72 x^{7} + 408 x^{6} - 297 x^{5} - 474 x^{4} + 616 x^{3} - 72 x^{2} - 144 x + 32 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1731826804433353307136=2^{10}\cdot 3^{15}\cdot 4903^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.06$
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:  $2, 3, 4903$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{46} a^{11} + \frac{9}{23} a^{10} - \frac{11}{23} a^{9} + \frac{9}{23} a^{8} - \frac{15}{46} a^{7} + \frac{9}{23} a^{5} + \frac{2}{23} a^{4} + \frac{9}{23} a^{3} + \frac{11}{23} a^{2} + \frac{13}{46} a - \frac{11}{23}$, $\frac{1}{6348} a^{12} + \frac{4}{529} a^{11} + \frac{255}{1058} a^{10} + \frac{1289}{3174} a^{9} + \frac{773}{2116} a^{8} - \frac{72}{529} a^{7} + \frac{174}{529} a^{6} - \frac{377}{1058} a^{5} - \frac{5}{23} a^{4} - \frac{233}{529} a^{3} + \frac{485}{2116} a^{2} - \frac{11}{46} a + \frac{778}{1587}$, $\frac{1}{292008} a^{13} - \frac{7}{146004} a^{12} - \frac{241}{48668} a^{11} - \frac{27097}{146004} a^{10} - \frac{87689}{292008} a^{9} - \frac{8237}{48668} a^{8} + \frac{3906}{12167} a^{7} + \frac{19309}{48668} a^{6} + \frac{5753}{24334} a^{5} - \frac{4751}{12167} a^{4} - \frac{30603}{97336} a^{3} + \frac{5581}{24334} a^{2} + \frac{11611}{73002} a - \frac{16183}{36501}$, $\frac{1}{13432368} a^{14} + \frac{1}{839523} a^{13} + \frac{125}{6716184} a^{12} + \frac{1997}{6716184} a^{11} + \frac{63931}{13432368} a^{10} + \frac{255727}{3358092} a^{9} + \frac{122253}{559682} a^{8} + \frac{1108017}{2238728} a^{7} - \frac{45391}{559682} a^{6} - \frac{166557}{559682} a^{5} + \frac{1067885}{4477456} a^{4} - \frac{411911}{2238728} a^{3} + \frac{94283}{1679046} a^{2} - \frac{170527}{1679046} a + \frac{314821}{839523}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $10$
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:  \( 96504.3069418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T93:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 933120
The 108 conjugacy class representatives for [S(3)^5]S(5)=S(3)wrS(5) are not computed
Character table for [S(3)^5]S(5)=S(3)wrS(5) is not computed

Intermediate fields

5.3.4903.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 45 sibling: 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 R R ${\href{/LocalNumberField/5.9.0.1}{9} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
$3$3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
3.9.9.9$x^{9} + 18 x^{5} + 27 x^{2} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
4903Data not computed