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Properties

Label 15.7.72891804143...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{23}\cdot 5^{7}\cdot 23^{4}\cdot 73^{7}\cdot 59981^{2}$
Root discriminant $454.48$
Ramified primes $2, 5, 23, 73, 59981$
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![-184000, -3220000, -20741400, -60970700, -83106360, -39366720, 13797194, 14691665, -359352, -1181655, 117274, 12096, -1980, -63, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 63*x^13 - 1980*x^12 + 12096*x^11 + 117274*x^10 - 1181655*x^9 - 359352*x^8 + 14691665*x^7 + 13797194*x^6 - 39366720*x^5 - 83106360*x^4 - 60970700*x^3 - 20741400*x^2 - 3220000*x - 184000)
 
gp: K = bnfinit(x^15 - 63*x^13 - 1980*x^12 + 12096*x^11 + 117274*x^10 - 1181655*x^9 - 359352*x^8 + 14691665*x^7 + 13797194*x^6 - 39366720*x^5 - 83106360*x^4 - 60970700*x^3 - 20741400*x^2 - 3220000*x - 184000, 1)
 

Normalized defining polynomial

\( x^{15} - 63 x^{13} - 1980 x^{12} + 12096 x^{11} + 117274 x^{10} - 1181655 x^{9} - 359352 x^{8} + 14691665 x^{7} + 13797194 x^{6} - 39366720 x^{5} - 83106360 x^{4} - 60970700 x^{3} - 20741400 x^{2} - 3220000 x - 184000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7289180414320769222686507858001920000000=2^{23}\cdot 5^{7}\cdot 23^{4}\cdot 73^{7}\cdot 59981^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $454.48$
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:  $2, 5, 23, 73, 59981$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{8} + \frac{3}{40} a^{7} - \frac{1}{20} a^{6} - \frac{9}{40} a^{5} - \frac{17}{40} a^{4} + \frac{9}{20} a^{3} - \frac{7}{20} a^{2}$, $\frac{1}{40} a^{9} - \frac{1}{40} a^{7} + \frac{7}{40} a^{6} - \frac{1}{4} a^{5} + \frac{19}{40} a^{4} - \frac{9}{20} a^{3} - \frac{9}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{40} a^{10} - \frac{1}{20} a^{6} - \frac{1}{4} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{7}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{40} a^{11} - \frac{1}{20} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{7}{20} a^{3}$, $\frac{1}{400} a^{12} + \frac{1}{400} a^{11} - \frac{1}{200} a^{10} - \frac{1}{200} a^{9} + \frac{1}{100} a^{8} - \frac{3}{100} a^{7} + \frac{83}{400} a^{6} - \frac{29}{400} a^{5} - \frac{77}{200} a^{4} - \frac{9}{40} a^{3}$, $\frac{1}{400} a^{13} - \frac{3}{400} a^{11} - \frac{1}{100} a^{9} + \frac{1}{100} a^{8} - \frac{7}{80} a^{7} - \frac{11}{200} a^{6} - \frac{1}{80} a^{5} + \frac{67}{200} a^{4} + \frac{3}{40} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{507417795487919884058605673376702271564000} a^{14} + \frac{9881526174506482814071884911943659451}{10148355909758397681172113467534045431280} a^{13} - \frac{602483997846040738901912129331496065533}{507417795487919884058605673376702271564000} a^{12} - \frac{87348730365849750719507934070340124019}{50741779548791988405860567337670227156400} a^{11} - \frac{1320775844009647494237010518336271463427}{253708897743959942029302836688351135782000} a^{10} + \frac{1297984371164821423837603601684835659397}{253708897743959942029302836688351135782000} a^{9} + \frac{240244069801331548766281238742367832089}{101483559097583976811721134675340454312800} a^{8} - \frac{26949713507179782287365837258169369847311}{253708897743959942029302836688351135782000} a^{7} + \frac{24499907743237665694935445719370118785747}{101483559097583976811721134675340454312800} a^{6} + \frac{13619533920387196562432055953564624407801}{126854448871979971014651418344175567891000} a^{5} + \frac{18179500020816463293529165981100333515467}{50741779548791988405860567337670227156400} a^{4} - \frac{580881602364289860271503928414069281201}{2537088977439599420293028366883511357820} a^{3} + \frac{313495143823112048185218650943397803937}{2537088977439599420293028366883511357820} a^{2} + \frac{20098311579650650691338087991437643367}{634272244359899855073257091720877839455} a + \frac{7388083622098324733755636970162872076}{126854448871979971014651418344175567891}$

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:  $10$
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:  \( 6422859770650000 \) (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

3.3.2920.1

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 ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.10.1$x^{4} + 2 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{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}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.5.4.1$x^{5} - 23$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.6.3.2$x^{6} - 5329 x^{2} + 5446238$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
59981Data not computed