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Properties

Label 15.3.50055120374...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{24}\cdot 5^{6}\cdot 17^{4}\cdot 29\cdot 37^{5}\cdot 311^{2}\cdot 342841^{2}$
Root discriminant $602.51$
Ramified primes $2, 5, 17, 29, 37, 311, 342841$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2367488000, 4143104000, 858214400, -3813875200, 592611840, 804741760, -280438208, 7178496, 5932512, -1330002, 67760, 32968, -2736, -486, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 486*x^13 - 2736*x^12 + 32968*x^11 + 67760*x^10 - 1330002*x^9 + 5932512*x^8 + 7178496*x^7 - 280438208*x^6 + 804741760*x^5 + 592611840*x^4 - 3813875200*x^3 + 858214400*x^2 + 4143104000*x - 2367488000)
 
gp: K = bnfinit(x^15 - 486*x^13 - 2736*x^12 + 32968*x^11 + 67760*x^10 - 1330002*x^9 + 5932512*x^8 + 7178496*x^7 - 280438208*x^6 + 804741760*x^5 + 592611840*x^4 - 3813875200*x^3 + 858214400*x^2 + 4143104000*x - 2367488000, 1)
 

Normalized defining polynomial

\( x^{15} - 486 x^{13} - 2736 x^{12} + 32968 x^{11} + 67760 x^{10} - 1330002 x^{9} + 5932512 x^{8} + 7178496 x^{7} - 280438208 x^{6} + 804741760 x^{5} + 592611840 x^{4} - 3813875200 x^{3} + 858214400 x^{2} + 4143104000 x - 2367488000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(500551203745655641831493145752502272000000=2^{24}\cdot 5^{6}\cdot 17^{4}\cdot 29\cdot 37^{5}\cdot 311^{2}\cdot 342841^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $602.51$
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:  $2, 5, 17, 29, 37, 311, 342841$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{9} + \frac{1}{16} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{7}{16} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{10} + \frac{1}{32} a^{8} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{9}{32} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1280} a^{11} - \frac{3}{640} a^{9} + \frac{1}{20} a^{8} + \frac{1}{160} a^{7} - \frac{3}{16} a^{6} - \frac{41}{640} a^{5} + \frac{11}{40} a^{4} + \frac{9}{20} a^{3} + \frac{11}{40} a^{2} - \frac{1}{2} a$, $\frac{1}{2560} a^{12} - \frac{3}{1280} a^{10} - \frac{1}{160} a^{9} + \frac{1}{320} a^{8} + \frac{3}{32} a^{7} - \frac{41}{1280} a^{6} - \frac{9}{80} a^{5} - \frac{11}{40} a^{4} - \frac{3}{10} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{870400} a^{13} + \frac{97}{435200} a^{11} - \frac{171}{54400} a^{10} + \frac{211}{108800} a^{9} + \frac{31}{10880} a^{8} - \frac{36681}{435200} a^{7} - \frac{3309}{27200} a^{6} + \frac{23571}{54400} a^{5} - \frac{5707}{13600} a^{4} - \frac{287}{680} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{62217271925548256763090608452534566530350227251200} a^{14} - \frac{2703587001711411192517417440360602669408201}{7777158990693532095386326056566820816293778406400} a^{13} + \frac{278749683350623261593889535196430858784913077}{31108635962774128381545304226267283265175113625600} a^{12} + \frac{63813294843096874081475094976442246311823113}{972144873836691511923290757070852602036722300800} a^{11} + \frac{23082225691561155806579894430231462538284864877}{7777158990693532095386326056566820816293778406400} a^{10} + \frac{17762012381575401709046468006272287082933213631}{3888579495346766047693163028283410408146889203200} a^{9} - \frac{978481408514195408273741301162624272933733230121}{31108635962774128381545304226267283265175113625600} a^{8} + \frac{3247694865266136203757005903763665286745419423}{3888579495346766047693163028283410408146889203200} a^{7} + \frac{70567787509323880459702096161288650426219467287}{388857949534676604769316302828341040814688920320} a^{6} + \frac{52223731509440759389622569768665598382567643471}{972144873836691511923290757070852602036722300800} a^{5} - \frac{163454210832799773291629233490808340994001604667}{486072436918345755961645378535426301018361150400} a^{4} + \frac{197608880743692776045781820387768258206350597}{24303621845917287798082268926771315050918057520} a^{3} + \frac{589878421363281915243521433045590061347405337}{1429624814465722811651898172163018532406944560} a^{2} - \frac{10447766605706237141265258093655087112345999}{35740620361643070291297454304075463310173614} a - \frac{6510989125941068707101261390681940475959818}{17870310180821535145648727152037731655086807}$

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:  $8$
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:  \( 10878315423800000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.6$x^{6} + 6 x^{4} + 6$$6$$1$$11$$S_4\times C_2$$[4/3, 4/3, 3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$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$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$17.5.4.1$x^{5} - 17$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
17.10.0.1$x^{10} - x + 7$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
311Data not computed
342841Data not computed