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Properties

Label 15.15.1341587352...1952.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{24}\cdot 3^{6}\cdot 257^{5}\cdot 31279007^{2}$
Root discriminant $298.66$
Ramified primes $2, 3, 257, 31279007$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T74

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![934107, 5570746, -16200387, -16552753, 18765059, 6160836, -6610445, -546699, 936068, -23098, -58872, 4894, 1475, -176, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 176*x^13 + 1475*x^12 + 4894*x^11 - 58872*x^10 - 23098*x^9 + 936068*x^8 - 546699*x^7 - 6610445*x^6 + 6160836*x^5 + 18765059*x^4 - 16552753*x^3 - 16200387*x^2 + 5570746*x + 934107)
 
gp: K = bnfinit(x^15 - 7*x^14 - 176*x^13 + 1475*x^12 + 4894*x^11 - 58872*x^10 - 23098*x^9 + 936068*x^8 - 546699*x^7 - 6610445*x^6 + 6160836*x^5 + 18765059*x^4 - 16552753*x^3 - 16200387*x^2 + 5570746*x + 934107, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 176 x^{13} + 1475 x^{12} + 4894 x^{11} - 58872 x^{10} - 23098 x^{9} + 936068 x^{8} - 546699 x^{7} - 6610445 x^{6} + 6160836 x^{5} + 18765059 x^{4} - 16552753 x^{3} - 16200387 x^{2} + 5570746 x + 934107 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13415873525861431436750950512514301952=2^{24}\cdot 3^{6}\cdot 257^{5}\cdot 31279007^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $298.66$
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:  $2, 3, 257, 31279007$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{83326291221025884147819678578406523090551421675} a^{14} + \frac{5296571960866306128293338991290820204906485399}{16665258244205176829563935715681304618110284335} a^{13} - \frac{38813569045962374346037334693338605207638062511}{83326291221025884147819678578406523090551421675} a^{12} - \frac{23463349935200790101823476779533187163073070047}{83326291221025884147819678578406523090551421675} a^{11} + \frac{902495699935253854892042524176595309432853582}{3333051648841035365912787143136260923622056867} a^{10} - \frac{31919487219942092169197697760627899183201837372}{83326291221025884147819678578406523090551421675} a^{9} + \frac{32290237885198324552090242411682781607197519558}{83326291221025884147819678578406523090551421675} a^{8} + \frac{8406930558629298392901020095134854012621978384}{83326291221025884147819678578406523090551421675} a^{7} - \frac{26339810221453000845248203627512504253371207631}{83326291221025884147819678578406523090551421675} a^{6} - \frac{4385257249189478012773465860444576736825980651}{11903755888717983449688525511200931870078774525} a^{5} - \frac{23079012557412443642564655049378355579733829453}{83326291221025884147819678578406523090551421675} a^{4} - \frac{8109203796919480838202271754571782602005789272}{83326291221025884147819678578406523090551421675} a^{3} - \frac{7832261888754316255208724211176709417923420847}{83326291221025884147819678578406523090551421675} a^{2} + \frac{2096435701149077868560498393162179636136606219}{83326291221025884147819678578406523090551421675} a - \frac{11702677693761272367746005024372105048336509416}{83326291221025884147819678578406523090551421675}$

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:  $14$
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:  \( 149313172695000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T74:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 40 conjugacy class representatives for [1/2.F(5)^3]S(3)
Character table for [1/2.F(5)^3]S(3) is not computed

Intermediate fields

3.3.257.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.24.225$x^{12} - 12 x^{11} + 16 x^{10} - 4 x^{9} - 10 x^{8} + 16 x^{7} - 8 x^{4} - 8 x^{2} + 8$$4$$3$$24$12T55$[2, 2, 3, 3]^{6}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
257Data not computed
31279007Data not computed