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Properties

Label 15.9.48986909510...7031.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,13^{5}\cdot 347^{3}\cdot 1777^{2}$
Root discriminant $20.54$
Ramified primes $13, 347, 1777$
Class number $1$
Class group Trivial
Galois group 15T78

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 172, -234, -298, 806, -409, -288, 465, -223, -19, 74, -33, 9, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 9*x^12 - 33*x^11 + 74*x^10 - 19*x^9 - 223*x^8 + 465*x^7 - 288*x^6 - 409*x^5 + 806*x^4 - 298*x^3 - 234*x^2 + 172*x - 19)
 
gp: K = bnfinit(x^15 - 3*x^14 + 9*x^12 - 33*x^11 + 74*x^10 - 19*x^9 - 223*x^8 + 465*x^7 - 288*x^6 - 409*x^5 + 806*x^4 - 298*x^3 - 234*x^2 + 172*x - 19, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} + 9 x^{12} - 33 x^{11} + 74 x^{10} - 19 x^{9} - 223 x^{8} + 465 x^{7} - 288 x^{6} - 409 x^{5} + 806 x^{4} - 298 x^{3} - 234 x^{2} + 172 x - 19 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-48986909510143987031=-\,13^{5}\cdot 347^{3}\cdot 1777^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.54$
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:  $13, 347, 1777$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{4}{9} a^{9} + \frac{2}{9} a^{7} + \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{3} a^{2} - \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{12} + \frac{5}{27} a^{10} - \frac{13}{27} a^{9} + \frac{2}{27} a^{8} - \frac{13}{27} a^{7} - \frac{13}{27} a^{6} + \frac{2}{9} a^{5} + \frac{4}{27} a^{4} - \frac{4}{9} a^{3} + \frac{13}{27} a^{2} - \frac{1}{3} a + \frac{2}{27}$, $\frac{1}{237242218113} a^{14} - \frac{1515381835}{237242218113} a^{13} + \frac{5114006770}{237242218113} a^{12} - \frac{25184557390}{237242218113} a^{11} + \frac{59510611780}{237242218113} a^{10} + \frac{6983471758}{237242218113} a^{9} - \frac{87262171529}{237242218113} a^{8} - \frac{55030630538}{237242218113} a^{7} - \frac{76354670854}{237242218113} a^{6} - \frac{93926944313}{237242218113} a^{5} - \frac{98984997806}{237242218113} a^{4} + \frac{31869441028}{237242218113} a^{3} + \frac{101601249568}{237242218113} a^{2} - \frac{76042207510}{237242218113} a - \frac{64116370333}{237242218113}$

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:  $11$
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:  \( 22178.4647209 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T78:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 29160
The 108 conjugacy class representatives for [3^5]S(5)=3wrS(5) are not computed
Character table for [3^5]S(5)=3wrS(5) is not computed

Intermediate fields

5.3.4511.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 siblings: 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 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
347Data not computed
1777Data not computed