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Properties

Label 15.7.44987474832...2288.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{20}\cdot 13^{4}\cdot 37^{5}\cdot 61^{3}\cdot 30893^{2}$
Root discriminant $150.27$
Ramified primes $2, 13, 37, 61, 30893$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-69121, 89497, 1035129, 1562899, 821871, 142641, -41575, -14857, 12185, 6975, 1683, -215, -187, -45, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 45*x^13 - 187*x^12 - 215*x^11 + 1683*x^10 + 6975*x^9 + 12185*x^8 - 14857*x^7 - 41575*x^6 + 142641*x^5 + 821871*x^4 + 1562899*x^3 + 1035129*x^2 + 89497*x - 69121)
 
gp: K = bnfinit(x^15 - x^14 - 45*x^13 - 187*x^12 - 215*x^11 + 1683*x^10 + 6975*x^9 + 12185*x^8 - 14857*x^7 - 41575*x^6 + 142641*x^5 + 821871*x^4 + 1562899*x^3 + 1035129*x^2 + 89497*x - 69121, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 45 x^{13} - 187 x^{12} - 215 x^{11} + 1683 x^{10} + 6975 x^{9} + 12185 x^{8} - 14857 x^{7} - 41575 x^{6} + 142641 x^{5} + 821871 x^{4} + 1562899 x^{3} + 1035129 x^{2} + 89497 x - 69121 \)

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:  \(449874748329512465067024908812288=2^{20}\cdot 13^{4}\cdot 37^{5}\cdot 61^{3}\cdot 30893^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $150.27$
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:  $2, 13, 37, 61, 30893$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{32} a^{13} + \frac{1}{32} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{5}{32} a^{9} + \frac{1}{32} a^{8} + \frac{1}{8} a^{7} - \frac{13}{32} a^{5} + \frac{15}{32} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{15}{32} a + \frac{15}{32}$, $\frac{1}{12071368451687063290058047496748306176} a^{14} - \frac{23660529038360176586618385040948639}{3017842112921765822514511874187076544} a^{13} - \frac{372887536410131667646627693797318681}{12071368451687063290058047496748306176} a^{12} + \frac{24637142483669353038814692000529481}{1508921056460882911257255937093538272} a^{11} + \frac{734950261753746064780417583314814801}{12071368451687063290058047496748306176} a^{10} + \frac{216884473109358745605538297308414557}{1508921056460882911257255937093538272} a^{9} - \frac{782002465002260616593787982214558329}{12071368451687063290058047496748306176} a^{8} - \frac{632698146325297269214386343650133985}{3017842112921765822514511874187076544} a^{7} - \frac{1257468671429463094247288612705893789}{12071368451687063290058047496748306176} a^{6} - \frac{265647750508011191460734744238675087}{1508921056460882911257255937093538272} a^{5} + \frac{5058376981277811441034289723233358297}{12071368451687063290058047496748306176} a^{4} + \frac{232024860417254278634475332020420107}{3017842112921765822514511874187076544} a^{3} - \frac{5435064215112767569820354406004721361}{12071368451687063290058047496748306176} a^{2} + \frac{753232085112855483949527141901953353}{3017842112921765822514511874187076544} a + \frac{4087264553200967628449792468282208269}{12071368451687063290058047496748306176}$

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:  \( 202085653591 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.18.66$x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{2} - 2$$12$$1$$18$12T98$[4/3, 4/3, 5/3, 5/3, 2]_{3}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.2$x^{8} + 169 x^{4} - 2197 x^{2} + 57122$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
37Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.8.0.1$x^{8} - x + 17$$1$$8$$0$$C_8$$[\ ]^{8}$
30893Data not computed