↑

Properties

Label 15.15.2682533404...4656.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 3^{20}\cdot 11^{10}\cdot 13^{2}\cdot 157^{6}\cdot 887^{2}\cdot 1907^{2}$
Root discriminant $2682.68$
Ramified primes $2, 3, 11, 13, 157, 887, 1907$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T77

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-148757485284891230462, -82917165854238062490, -17963813846668521546, -1756261687063060122, -42397281676187196, 6060790547069841, 471963727156998, 2179565290971, -850155338070, -21753227652, 494149740, 20821806, -91336, -7599, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7599*x^13 - 91336*x^12 + 20821806*x^11 + 494149740*x^10 - 21753227652*x^9 - 850155338070*x^8 + 2179565290971*x^7 + 471963727156998*x^6 + 6060790547069841*x^5 - 42397281676187196*x^4 - 1756261687063060122*x^3 - 17963813846668521546*x^2 - 82917165854238062490*x - 148757485284891230462)
 
gp: K = bnfinit(x^15 - 7599*x^13 - 91336*x^12 + 20821806*x^11 + 494149740*x^10 - 21753227652*x^9 - 850155338070*x^8 + 2179565290971*x^7 + 471963727156998*x^6 + 6060790547069841*x^5 - 42397281676187196*x^4 - 1756261687063060122*x^3 - 17963813846668521546*x^2 - 82917165854238062490*x - 148757485284891230462, 1)
 

Normalized defining polynomial

\( x^{15} - 7599 x^{13} - 91336 x^{12} + 20821806 x^{11} + 494149740 x^{10} - 21753227652 x^{9} - 850155338070 x^{8} + 2179565290971 x^{7} + 471963727156998 x^{6} + 6060790547069841 x^{5} - 42397281676187196 x^{4} - 1756261687063060122 x^{3} - 17963813846668521546 x^{2} - 82917165854238062490 x - 148757485284891230462 \)

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:  \(2682533404519714891839773452435223413236182431174656=2^{12}\cdot 3^{20}\cdot 11^{10}\cdot 13^{2}\cdot 157^{6}\cdot 887^{2}\cdot 1907^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2682.68$
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:  $2, 3, 11, 13, 157, 887, 1907$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{22} a^{12} + \frac{1}{11} a^{10} - \frac{3}{22} a^{9} + \frac{5}{22} a^{8} - \frac{2}{11} a^{7} - \frac{1}{11} a^{6} - \frac{5}{22} a^{5} + \frac{1}{11} a^{4} + \frac{5}{11} a^{3}$, $\frac{1}{22} a^{13} + \frac{1}{11} a^{11} - \frac{3}{22} a^{10} + \frac{5}{22} a^{9} - \frac{2}{11} a^{8} - \frac{1}{11} a^{7} - \frac{5}{22} a^{6} + \frac{1}{11} a^{5} + \frac{5}{11} a^{4}$, $\frac{1}{413343611468805491099251520461931184550453525589206788569891891687446678023180772455793328391219545493541117955548678} a^{14} + \frac{624126028604502425906131217947021478699361806572728808522685069898324319281161823345641294151896522846118992917895}{37576691951709590099931956405630107686404865962655162597262899244313334365743706586890302581019958681231010723231698} a^{13} - \frac{7376232012598583664989469971362708256482215986983162570682582706929790654092724357946207793777837806406097145677619}{413343611468805491099251520461931184550453525589206788569891891687446678023180772455793328391219545493541117955548678} a^{12} + \frac{27901704925556578639188140282850565717693702399726013949792961036928303522092629554953284077045164664628396065830016}{206671805734402745549625760230965592275226762794603394284945945843723339011590386227896664195609772746770558977774339} a^{11} - \frac{102331233867505739146687056714166859566294854005399537637941604949738788873512511126527283837187524253934299369327155}{413343611468805491099251520461931184550453525589206788569891891687446678023180772455793328391219545493541117955548678} a^{10} - \frac{7945941450821492605271099190777941384032025468220223637561329704054387601717481302842524299446189966158103860002662}{18788345975854795049965978202815053843202432981327581298631449622156667182871853293445151290509979340615505361615849} a^{9} - \frac{172823915918568233632780379762242910519755003090385192982146254666726130897387843617835619578214058250767717499070911}{413343611468805491099251520461931184550453525589206788569891891687446678023180772455793328391219545493541117955548678} a^{8} - \frac{86648224303297168335879414403109947673738971812975767724553104902545080997072499673693884730176431391910244142508463}{206671805734402745549625760230965592275226762794603394284945945843723339011590386227896664195609772746770558977774339} a^{7} + \frac{68488725366200724590238009804358306796807310242361930171459222633530459302399703082976008853054425746387154782091097}{206671805734402745549625760230965592275226762794603394284945945843723339011590386227896664195609772746770558977774339} a^{6} + \frac{9274676822055762941405300961974036335713021242647160575155794732696767872909935332627245910146681089218008969827751}{413343611468805491099251520461931184550453525589206788569891891687446678023180772455793328391219545493541117955548678} a^{5} + \frac{23925086510906142578865813784534741900515396886605126392786173563467475066237413769220509560026205518042405580990729}{206671805734402745549625760230965592275226762794603394284945945843723339011590386227896664195609772746770558977774339} a^{4} - \frac{61426583641362109176607638382401908941104068890256535309471297709474463371027715276087275128740391056296156156470812}{206671805734402745549625760230965592275226762794603394284945945843723339011590386227896664195609772746770558977774339} a^{3} - \frac{8905192719614326461477107184951221409375838394477567269201060068644121592184922431486818391681553476969856829921881}{18788345975854795049965978202815053843202432981327581298631449622156667182871853293445151290509979340615505361615849} a^{2} - \frac{3512876844984275312846027939244022443407258244666609102727546525707862978808642019875819010111500631191250439155575}{18788345975854795049965978202815053843202432981327581298631449622156667182871853293445151290509979340615505361615849} a + \frac{259229187008908848472793853693858856511051339383712279761284548686139012388348727203310852376300181648358980505133}{18788345975854795049965978202815053843202432981327581298631449622156667182871853293445151290509979340615505361615849}$

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:  \( 1702104381240000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T77:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 29160
The 48 conjugacy class representatives for 1/2[3^5:2]S(5)
Character table for 1/2[3^5:2]S(5) is not computed

Intermediate fields

5.5.303952.1

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
Degree 45 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ R R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3Data not computed
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.9.6.1$x^{9} - 121 x^{3} + 3993$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$157$157.3.0.1$x^{3} - x + 15$$1$$3$$0$$C_3$$[\ ]^{3}$
157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
887Data not computed
1907Data not computed