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Properties

Label 15.1.22909594796...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 3^{20}\cdot 5^{18}\cdot 1459^{8}$
Root discriminant $4207.28$
Ramified primes $2, 3, 5, 1459$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group 15T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6241061091507200, -5044667439936000, -1765231283268600, -265590632403700, -2532704653800, 3532946629590, -85198013200, -83929544010, -2590483680, 332396675, 1914312, -157785, -5640, 105, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 105*x^13 - 5640*x^12 - 157785*x^11 + 1914312*x^10 + 332396675*x^9 - 2590483680*x^8 - 83929544010*x^7 - 85198013200*x^6 + 3532946629590*x^5 - 2532704653800*x^4 - 265590632403700*x^3 - 1765231283268600*x^2 - 5044667439936000*x - 6241061091507200)
 
gp: K = bnfinit(x^15 + 105*x^13 - 5640*x^12 - 157785*x^11 + 1914312*x^10 + 332396675*x^9 - 2590483680*x^8 - 83929544010*x^7 - 85198013200*x^6 + 3532946629590*x^5 - 2532704653800*x^4 - 265590632403700*x^3 - 1765231283268600*x^2 - 5044667439936000*x - 6241061091507200, 1)
 

Normalized defining polynomial

\( x^{15} + 105 x^{13} - 5640 x^{12} - 157785 x^{11} + 1914312 x^{10} + 332396675 x^{9} - 2590483680 x^{8} - 83929544010 x^{7} - 85198013200 x^{6} + 3532946629590 x^{5} - 2532704653800 x^{4} - 265590632403700 x^{3} - 1765231283268600 x^{2} - 5044667439936000 x - 6241061091507200 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2290959479641121893981744591147824672000000000000000000=-\,2^{23}\cdot 3^{20}\cdot 5^{18}\cdot 1459^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $4207.28$
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:  $2, 3, 5, 1459$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{14590} a^{10} + \frac{21}{2918} a^{8} - \frac{564}{1459} a^{7} + \frac{541}{2918} a^{6} + \frac{1511}{7295} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{14590} a^{11} + \frac{21}{2918} a^{9} + \frac{331}{2918} a^{8} + \frac{541}{2918} a^{7} + \frac{1511}{7295} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{29180} a^{12} - \frac{1}{29180} a^{10} - \frac{282}{1459} a^{9} + \frac{1233}{5836} a^{8} + \frac{668}{7295} a^{7} + \frac{2473}{5836} a^{6} + \frac{162}{7295} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{58360} a^{13} + \frac{1}{58360} a^{11} + \frac{1275}{11672} a^{9} - \frac{1}{7295} a^{8} - \frac{2961}{11672} a^{7} + \frac{219}{7295} a^{6} + \frac{1755}{5836} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{158889809863541317439563268599342967697949472290082502140716820608065703939964733072320} a^{14} + \frac{15803566120208546617583507569956412106396521140137260467847923897216903301357895}{1986122623294266467994540857491787096224368403626031276758960257600821299249559163404} a^{13} - \frac{21332784675010656337187263958155202228419535618403122915898433576318010868020169}{5125477737533590885147202212882031216062886202905887165829574858324700127095636550720} a^{12} - \frac{121119738823627330449003692337351121353946243815118474177133072626835563886722599}{3972245246588532935989081714983574192448736807252062553517920515201642598499118326808} a^{11} - \frac{3222894105136239274532073927903832969148181041892083861081795453729796769141679193}{158889809863541317439563268599342967697949472290082502140716820608065703939964733072320} a^{10} - \frac{1035507217746382031134371055844373356256173508339313503432230133383400721063274323901}{19861226232942664679945408574917870962243684036260312767589602576008212992495591634040} a^{9} - \frac{5896708112075190411407812740667238595235772009900378314401394203690690506233200134649}{31777961972708263487912653719868593539589894458016500428143364121613140787992946614464} a^{8} + \frac{4637808523940180314870737130356287483100554216422110329917568369113197672205790373447}{9930613116471332339972704287458935481121842018130156383794801288004106496247795817020} a^{7} - \frac{1856755254178117226789602260900158510438085836585446931237373053275920794527232454497}{15888980986354131743956326859934296769794947229008250214071682060806570393996473307232} a^{6} - \frac{702615812900238343192349680712960662144185920203099404002469270168441344320737933233}{1418659016638761762853243469636990783017406002590022340542114469714872356606827973860} a^{5} + \frac{736972815536469737725837675668148967980786096865906404565794455159779047272210163}{1555760402071294599427820117490874059511891435328331559196287286870319239596247264} a^{4} + \frac{1078642800727434618549147180187994933367341320554927498331474502106095989376094065}{2722580703624765548998685205609029604145810011824580228593502752023058669293432712} a^{3} - \frac{615745414273204996720964856141862067422814684929996837592718926023175547493759139}{5445161407249531097997370411218059208291620023649160457187005504046117338586865424} a^{2} + \frac{1136673824762350521098531121943243309330046319385864673175796360923885713545243375}{2722580703624765548998685205609029604145810011824580228593502752023058669293432712} a + \frac{2823315271500987507275450923766672113803451506595303987411210438723234360412641}{48617512564727956232119378671589814359746607354010361224883977714697476237382727}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  $7$
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:  \( 6395084156450000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 40 conjugacy class representatives for [5^3:2]S(3)
Character table for [5^3:2]S(3) is not computed

Intermediate fields

3.1.648.1

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

Sibling fields

Degree 15 siblings: 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 R R R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
3Data not computed
$5$5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.10.12.10$x^{10} + 10 x^{8} + 20 x^{7} + 15 x^{6} - 5 x^{5} + 5 x^{4} + 5 x^{2} - 5 x + 7$$5$$2$$12$$D_{10}$$[3/2]_{2}^{2}$
1459Data not computed