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Properties

Label 15.5.19716834062...0000.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{16}\cdot 5^{6}\cdot 17^{4}\cdot 41^{2}\cdot 431^{5}\cdot 3036643^{2}$
Root discriminant $769.70$
Ramified primes $2, 5, 17, 41, 431, 3036643$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T100

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2228224000, -5849088000, -5918720000, -5079654400, -3865098240, -1490235840, -766844496, -169764568, -14595012, -710082, 463062, 46185, -3204, -297, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 297*x^13 - 3204*x^12 + 46185*x^11 + 463062*x^10 - 710082*x^9 - 14595012*x^8 - 169764568*x^7 - 766844496*x^6 - 1490235840*x^5 - 3865098240*x^4 - 5079654400*x^3 - 5918720000*x^2 - 5849088000*x - 2228224000)
 
gp: K = bnfinit(x^15 - 297*x^13 - 3204*x^12 + 46185*x^11 + 463062*x^10 - 710082*x^9 - 14595012*x^8 - 169764568*x^7 - 766844496*x^6 - 1490235840*x^5 - 3865098240*x^4 - 5079654400*x^3 - 5918720000*x^2 - 5849088000*x - 2228224000, 1)
 

Normalized defining polynomial

\( x^{15} - 297 x^{13} - 3204 x^{12} + 46185 x^{11} + 463062 x^{10} - 710082 x^{9} - 14595012 x^{8} - 169764568 x^{7} - 766844496 x^{6} - 1490235840 x^{5} - 3865098240 x^{4} - 5079654400 x^{3} - 5918720000 x^{2} - 5849088000 x - 2228224000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-19716834062238531710965065164233726976000000=-\,2^{16}\cdot 5^{6}\cdot 17^{4}\cdot 41^{2}\cdot 431^{5}\cdot 3036643^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $769.70$
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:  $2, 5, 17, 41, 431, 3036643$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{8} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{9} + \frac{3}{32} a^{7} - \frac{1}{8} a^{6} + \frac{13}{32} a^{5} + \frac{3}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{128} a^{10} - \frac{1}{64} a^{9} + \frac{3}{128} a^{8} + \frac{3}{64} a^{7} + \frac{21}{128} a^{6} + \frac{15}{32} a^{5} - \frac{25}{64} a^{4} - \frac{1}{4} a^{3} + \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{1280} a^{11} - \frac{17}{1280} a^{9} - \frac{1}{320} a^{8} - \frac{3}{256} a^{7} - \frac{69}{640} a^{6} - \frac{101}{640} a^{5} - \frac{13}{320} a^{4} + \frac{59}{160} a^{3} - \frac{31}{80} a^{2} - \frac{1}{4} a$, $\frac{1}{10240} a^{12} + \frac{23}{10240} a^{10} - \frac{1}{2560} a^{9} - \frac{43}{2048} a^{8} + \frac{171}{5120} a^{7} - \frac{321}{5120} a^{6} + \frac{1167}{2560} a^{5} - \frac{11}{1280} a^{4} + \frac{219}{640} a^{3} + \frac{5}{32} a^{2} - \frac{1}{2} a$, $\frac{1}{409600} a^{13} + \frac{23}{409600} a^{11} - \frac{1}{102400} a^{10} - \frac{811}{81920} a^{9} + \frac{5291}{204800} a^{8} + \frac{4159}{204800} a^{7} - \frac{7793}{102400} a^{6} + \frac{6549}{51200} a^{5} - \frac{7621}{25600} a^{4} - \frac{87}{256} a^{3} + \frac{2}{5} a^{2} + \frac{1}{4} a$, $\frac{1}{286345046873752895691795163872936287445998925640499200} a^{14} + \frac{42995370258927836388339878809490044136144814919}{35793130859219111961474395484117035930749865705062400} a^{13} + \frac{3074261176731833757797524666376400196461397719063}{286345046873752895691795163872936287445998925640499200} a^{12} - \frac{2307298215875737788960713813868826136673514805481}{10226608816919746274706970138319153123071390201446400} a^{11} + \frac{544840969765852240161195806187370432645615966634057}{286345046873752895691795163872936287445998925640499200} a^{10} + \frac{282390114787234571278968144095166173004857820318113}{20453217633839492549413940276638306246142780402892800} a^{9} + \frac{1935769873768779179433580785887640618521079960950311}{143172523436876447845897581936468143722999462820249600} a^{8} + \frac{4946927921603462795601381453295108482611855311827411}{71586261718438223922948790968234071861499731410124800} a^{7} - \frac{631552316535619539121450310549103384785688478301319}{35793130859219111961474395484117035930749865705062400} a^{6} - \frac{801745747258700161064374948644154864540007560100751}{2556652204229936568676742534579788280767847550361600} a^{5} + \frac{150050291083409981004903262398805768369569011501627}{4474141357402388995184299435514629491343733213132800} a^{4} + \frac{48261611553555282429521386285273698597848082706693}{111853533935059724879607485887865737283593330328320} a^{3} + \frac{1240980069295067183700370251907730271910884571923}{13981691741882465609950935735983217160449166291040} a^{2} - \frac{19904507790688885098286843122810510627877036750}{87385573386765410062193348349895107252807289319} a - \frac{31090924930160347949581558816387318543170197809}{87385573386765410062193348349895107252807289319}$

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:  $9$
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:  \( 547642073776000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T100:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed
Character table for [1/2.S(5)^3]S(3) is not computed

Intermediate fields

3.1.431.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ R ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.5.4.1$x^{5} - 17$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
41Data not computed
431Data not computed
3036643Data not computed