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Properties

Label 15.7.14902168363...5625.1
Degree $15$
Signature $[7, 4]$
Discriminant $5^{4}\cdot 13^{10}\cdot 331^{12}$
Root discriminant $880.81$
Ramified primes $5, 13, 331$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group 15T39

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7167576473465, 17219135199695, 12917190317191, 3695380376144, 688125232186, 44995084425, -13883866784, -1027018697, 137318238, -7366320, -1031495, 126567, 4156, -640, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 640*x^13 + 4156*x^12 + 126567*x^11 - 1031495*x^10 - 7366320*x^9 + 137318238*x^8 - 1027018697*x^7 - 13883866784*x^6 + 44995084425*x^5 + 688125232186*x^4 + 3695380376144*x^3 + 12917190317191*x^2 + 17219135199695*x + 7167576473465)
 
gp: K = bnfinit(x^15 - 7*x^14 - 640*x^13 + 4156*x^12 + 126567*x^11 - 1031495*x^10 - 7366320*x^9 + 137318238*x^8 - 1027018697*x^7 - 13883866784*x^6 + 44995084425*x^5 + 688125232186*x^4 + 3695380376144*x^3 + 12917190317191*x^2 + 17219135199695*x + 7167576473465, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 640 x^{13} + 4156 x^{12} + 126567 x^{11} - 1031495 x^{10} - 7366320 x^{9} + 137318238 x^{8} - 1027018697 x^{7} - 13883866784 x^{6} + 44995084425 x^{5} + 688125232186 x^{4} + 3695380376144 x^{3} + 12917190317191 x^{2} + 17219135199695 x + 7167576473465 \)

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:  \(149021683637623372924986597528084838454055625=5^{4}\cdot 13^{10}\cdot 331^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $880.81$
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:  $5, 13, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{35} a^{11} + \frac{2}{35} a^{10} - \frac{9}{35} a^{9} - \frac{3}{35} a^{8} + \frac{3}{35} a^{7} - \frac{3}{7} a^{6} - \frac{11}{35} a^{5} - \frac{2}{35} a^{4} - \frac{9}{35} a^{3} - \frac{2}{35} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{35} a^{12} + \frac{1}{35} a^{10} - \frac{6}{35} a^{9} + \frac{9}{35} a^{8} + \frac{1}{5} a^{7} - \frac{2}{35} a^{6} + \frac{6}{35} a^{5} - \frac{1}{7} a^{4} - \frac{12}{35} a^{3} - \frac{4}{35} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{245} a^{13} + \frac{1}{245} a^{12} - \frac{3}{245} a^{11} + \frac{8}{245} a^{10} - \frac{2}{49} a^{9} - \frac{6}{35} a^{8} + \frac{1}{7} a^{7} - \frac{4}{49} a^{6} - \frac{11}{245} a^{5} - \frac{79}{245} a^{4} + \frac{48}{245} a^{3} + \frac{57}{245} a^{2} + \frac{12}{49} a - \frac{22}{49}$, $\frac{1}{1031008857357679070058651306626554197694909408654305970826992443219902602583012533698725} a^{14} + \frac{236169429304129999210886569914953141657942508728623949527222465103368338307697817902}{147286979622525581436950186660936313956415629807757995832427491888557514654716076242675} a^{13} + \frac{6015124517507623319978581327874627349107158675177903027483261085405438445743394211169}{1031008857357679070058651306626554197694909408654305970826992443219902602583012533698725} a^{12} + \frac{2942939290853943706172609515363040223918974905176297363534311434398608158669554190742}{206201771471535814011730261325310839538981881730861194165398488643980520516602506739745} a^{11} - \frac{32908809570586354299412070317217434429797077923007846356407403848179439780652512856973}{1031008857357679070058651306626554197694909408654305970826992443219902602583012533698725} a^{10} - \frac{509707010575987344918227046817925052098073715361224822042423536896695598846862270149278}{1031008857357679070058651306626554197694909408654305970826992443219902602583012533698725} a^{9} - \frac{71223270098556555314769550288244480468799779985241579156773646740307085290735566804079}{147286979622525581436950186660936313956415629807757995832427491888557514654716076242675} a^{8} - \frac{3552604732167216116326013690154168362559424502775858063161234719344713756086221618386}{206201771471535814011730261325310839538981881730861194165398488643980520516602506739745} a^{7} - \frac{421537281543751347125911311799203860595734077123199972458924254517631938789014900538562}{1031008857357679070058651306626554197694909408654305970826992443219902602583012533698725} a^{6} - \frac{51776871964013497674765404758836747985932619096987721271403614249421636116734111420141}{1031008857357679070058651306626554197694909408654305970826992443219902602583012533698725} a^{5} - \frac{292426754316426131348349578706008580582333986710154846467573253295629683718594586412606}{1031008857357679070058651306626554197694909408654305970826992443219902602583012533698725} a^{4} + \frac{57534927077774127163761287937214553032230135275514751710551687132780183929140902758647}{206201771471535814011730261325310839538981881730861194165398488643980520516602506739745} a^{3} - \frac{232801167460548515704869455570322344841371579307384246746717321013202278086908980725736}{1031008857357679070058651306626554197694909408654305970826992443219902602583012533698725} a^{2} + \frac{57847503251661755404580080014852286305487109299796788923481198833580002665681092127123}{206201771471535814011730261325310839538981881730861194165398488643980520516602506739745} a - \frac{31829683435658849628039800618862042650032535497799446547884251296053670493754125713233}{206201771471535814011730261325310839538981881730861194165398488643980520516602506739745}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 21070099870400000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T39:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 28 conjugacy class representatives for [1/2.D(5)^3]3
Character table for [1/2.D(5)^3]3 is not computed

Intermediate fields

3.3.169.1

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

Sibling fields

Degree 20 sibling: data not computed
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 $15$ $15$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ R $15$ $15$ $15$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $15$ $15$ $15$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $15$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
13Data not computed
331Data not computed