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Properties

Label 15.15.1914620951...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 29^{4}\cdot 41^{4}\cdot 287744824009^{2}$
Root discriminant $3565.65$
Ramified primes $2, 5, 7, 29, 41, 287744824009$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T92

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![542000152064000, 6375394044674816000, 1792435586109190400, 80444100448171200, -15834444242156160, -1236668673788960, 47814832667776, 5167148500176, -63850635200, -10043207752, 37878064, 10106636, -7752, -5082, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5082*x^13 - 7752*x^12 + 10106636*x^11 + 37878064*x^10 - 10043207752*x^9 - 63850635200*x^8 + 5167148500176*x^7 + 47814832667776*x^6 - 1236668673788960*x^5 - 15834444242156160*x^4 + 80444100448171200*x^3 + 1792435586109190400*x^2 + 6375394044674816000*x + 542000152064000)
 
gp: K = bnfinit(x^15 - 5082*x^13 - 7752*x^12 + 10106636*x^11 + 37878064*x^10 - 10043207752*x^9 - 63850635200*x^8 + 5167148500176*x^7 + 47814832667776*x^6 - 1236668673788960*x^5 - 15834444242156160*x^4 + 80444100448171200*x^3 + 1792435586109190400*x^2 + 6375394044674816000*x + 542000152064000, 1)
 

Normalized defining polynomial

\( x^{15} - 5082 x^{13} - 7752 x^{12} + 10106636 x^{11} + 37878064 x^{10} - 10043207752 x^{9} - 63850635200 x^{8} + 5167148500176 x^{7} + 47814832667776 x^{6} - 1236668673788960 x^{5} - 15834444242156160 x^{4} + 80444100448171200 x^{3} + 1792435586109190400 x^{2} + 6375394044674816000 x + 542000152064000 \)

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:  \(191462095148251089869153304845837565344604688384000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 29^{4}\cdot 41^{4}\cdot 287744824009^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3565.65$
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:  $2, 5, 7, 29, 41, 287744824009$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{160} a^{11} - \frac{1}{80} a^{9} - \frac{1}{80} a^{8} - \frac{1}{40} a^{7} + \frac{1}{40} a^{6} + \frac{1}{20} a^{5} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{320} a^{12} - \frac{1}{160} a^{10} + \frac{1}{40} a^{9} - \frac{1}{80} a^{8} - \frac{1}{20} a^{7} + \frac{1}{40} a^{6} + \frac{1}{20} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{19200} a^{13} + \frac{1}{960} a^{12} - \frac{7}{3200} a^{11} + \frac{13}{1200} a^{10} + \frac{79}{4800} a^{9} - \frac{31}{1200} a^{8} - \frac{29}{2400} a^{7} - \frac{1}{40} a^{6} - \frac{119}{1200} a^{5} - \frac{3}{25} a^{4} + \frac{1}{8} a^{3} - \frac{1}{10} a^{2} + \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{70222136887736675117281752567106631071698938077893625007735067595485251616777870403586688683596800} a^{14} + \frac{18756273951320986384060773886715926512909867191005263846222199188582398333092284662688663797}{877776711096708438966021907088832888396236725973670312596688344943565645209723380044833608544960} a^{13} - \frac{6311084390143651896991931675198879486519522100385392376710365028701758059224840909761096448941}{35111068443868337558640876283553315535849469038946812503867533797742625808388935201793344341798400} a^{12} + \frac{11662254990165951093478826977923190541202282055836838858082192380214990424871246169757616116731}{8777767110967084389660219070888328883962367259736703125966883449435656452097233800448336085449600} a^{11} - \frac{3610695317176033277389211977600828411372900634567066465940207187596407254596891590216814924007}{5851844740644722926440146047258885922641578173157802083977922299623770968064822533632224056966400} a^{10} - \frac{60700322599161580107612513902773761097244806587116080911242951891038719081756080508715525946801}{4388883555483542194830109535444164441981183629868351562983441724717828226048616900224168042724800} a^{9} - \frac{29680950624289237550230051297563945846688433040358615718045568129435713226640846738927197152241}{975307456774120487740024341209814320440263028859633680662987049937295161344137088938704009494400} a^{8} - \frac{953426342451567740932447982476912140553419009902665884130991167201581662784869130000685462961}{43888835554835421948301095354441644419811836298683515629834417247178282260486169002241680427248} a^{7} - \frac{212380999064415296324179714741555178014747423751317849865304684294718640858167642125449072478659}{4388883555483542194830109535444164441981183629868351562983441724717828226048616900224168042724800} a^{6} - \frac{32053030851546171762666901717423927504360505903014618918177693425691173464917991883678693373343}{548610444435442774353763691930520555247647953733543945372930215589728528256077112528021005340600} a^{5} + \frac{116956871355100590507115982661402215237774416175905384333318162603599409220904190052091396313}{1008938748387021194213818284010152745283030719509965876547917637866167408287038367867624837408} a^{4} + \frac{134720869020922189868587226286864775561690422062411041120523993957288809805890699453502871817}{1261173435483776492767272855012690931603788399387457345684897047332709260358797959834531046760} a^{3} + \frac{192802945820077328178717397848541552740713889356610816357145949936855814002589344759864357491}{840782290322517661844848570008460621069192266258304897123264698221806173572531973223020697840} a^{2} - \frac{173744290093790472623921492952178487614295087658468263365886779837513410174525361918829365943}{378352030645132947830181856503807279481136519816237203705469114199812778107639387950359314028} a + \frac{9784651319838805828155176439101415833524143705070664371809073440962836262356264848491749472}{94588007661283236957545464125951819870284129954059300926367278549953194526909846987589828507}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 3288787740280000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T92:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 648000
The 55 conjugacy class representatives for [A(5)^3]3=A(5)wr3 are not computed
Character table for [A(5)^3]3=A(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 sibling: data not computed
Degree 36 sibling: 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 R ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ R $15$ $15$ R ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$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}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
$41$41.5.4.5$x^{5} - 53136$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.0.1$x^{5} - x + 7$$1$$5$$0$$C_5$$[\ ]^{5}$
41.5.0.1$x^{5} - x + 7$$1$$5$$0$$C_5$$[\ ]^{5}$
287744824009Data not computed