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Properties

Label 15.7.65659847052...5625.1
Degree $15$
Signature $[7, 4]$
Discriminant $3^{20}\cdot 5^{16}\cdot 47^{6}\cdot 107^{2}$
Root discriminant $209.49$
Ramified primes $3, 5, 47, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T84

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5778000, -2889000, 8667000, 2781200, -5506575, -1980945, 863075, 367620, 38700, 23795, 1530, -2880, -215, 45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 45*x^13 - 215*x^12 - 2880*x^11 + 1530*x^10 + 23795*x^9 + 38700*x^8 + 367620*x^7 + 863075*x^6 - 1980945*x^5 - 5506575*x^4 + 2781200*x^3 + 8667000*x^2 - 2889000*x - 5778000)
 
gp: K = bnfinit(x^15 + 45*x^13 - 215*x^12 - 2880*x^11 + 1530*x^10 + 23795*x^9 + 38700*x^8 + 367620*x^7 + 863075*x^6 - 1980945*x^5 - 5506575*x^4 + 2781200*x^3 + 8667000*x^2 - 2889000*x - 5778000, 1)
 

Normalized defining polynomial

\( x^{15} + 45 x^{13} - 215 x^{12} - 2880 x^{11} + 1530 x^{10} + 23795 x^{9} + 38700 x^{8} + 367620 x^{7} + 863075 x^{6} - 1980945 x^{5} - 5506575 x^{4} + 2781200 x^{3} + 8667000 x^{2} - 2889000 x - 5778000 \)

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:  \(65659847052759660683307037353515625=3^{20}\cdot 5^{16}\cdot 47^{6}\cdot 107^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $209.49$
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:  $3, 5, 47, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{10} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{300} a^{13} - \frac{1}{20} a^{11} - \frac{1}{60} a^{10} - \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{11}{60} a^{7} + \frac{2}{5} a^{5} - \frac{1}{12} a^{4} - \frac{3}{20} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a$, $\frac{1}{148898145757508676577783620088050386685670200} a^{14} - \frac{12818567787596456717514931496776983514411}{24816357626251446096297270014675064447611700} a^{13} + \frac{131549711164273058500411218029318631170373}{3308847683500192812839636001956675259681560} a^{12} - \frac{1447661973823592108150192395838405881732549}{29779629151501735315556724017610077337134040} a^{11} - \frac{120628963827005939907423642196422255787121}{4963271525250289219259454002935012889522340} a^{10} - \frac{224843645917964648129015042036210238181261}{1654423841750096406419818000978337629840780} a^{9} + \frac{3735759296587845983055897279411514001712643}{29779629151501735315556724017610077337134040} a^{8} + \frac{2289471611980525893372092751844088835967721}{4963271525250289219259454002935012889522340} a^{7} - \frac{8354268654949287430350310734146670902914}{1240817881312572304814863500733753222380585} a^{6} + \frac{10313230019465038642059382538361278326683151}{29779629151501735315556724017610077337134040} a^{5} + \frac{4033368009748994722976167324194516397370047}{9926543050500578438518908005870025779044680} a^{4} - \frac{3663611114216002212921647296536700343736007}{9926543050500578438518908005870025779044680} a^{3} - \frac{1427472097766717326890540888298529477742143}{2977962915150173531555672401761007733713404} a^{2} - \frac{189698425067585201058861659169606290735611}{496327152525028921925945400293501288952234} a + \frac{17275987409321802099546975919038995573165}{82721192087504820320990900048916881492039}$

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:  \( 2005835231560 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T84:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 39 conjugacy class representatives for 1/2[S(3)^5]F(5)
Character table for 1/2[S(3)^5]F(5) is not computed

Intermediate fields

5.5.6903125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$3$3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.12.16.19$x^{12} + 66 x^{11} - 45 x^{10} - 120 x^{9} + 9 x^{8} + 108 x^{7} + 18 x^{6} - 108 x^{5} - 81 x^{4} - 54 x^{3} - 81 x^{2} - 81$$3$$4$$16$12T46$[2, 2]^{8}$
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.12.6.2$x^{12} - 229345007 x^{2} + 53896076645$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
107Data not computed