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Properties

Label 15.7.39319960333...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{23}\cdot 5^{6}\cdot 37^{5}\cdot 79^{4}\cdot 233^{2}\cdot 857^{2}\cdot 1669^{2}$
Root discriminant $805.95$
Ramified primes $2, 5, 37, 79, 233, 857, 1669$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T97

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10112000, 70784000, 171145600, 153260000, 4550400, -58165280, -16677328, 6356616, 2916576, -296010, -91568, 7942, -504, -270, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 270*x^13 - 504*x^12 + 7942*x^11 - 91568*x^10 - 296010*x^9 + 2916576*x^8 + 6356616*x^7 - 16677328*x^6 - 58165280*x^5 + 4550400*x^4 + 153260000*x^3 + 171145600*x^2 + 70784000*x + 10112000)
 
gp: K = bnfinit(x^15 - 270*x^13 - 504*x^12 + 7942*x^11 - 91568*x^10 - 296010*x^9 + 2916576*x^8 + 6356616*x^7 - 16677328*x^6 - 58165280*x^5 + 4550400*x^4 + 153260000*x^3 + 171145600*x^2 + 70784000*x + 10112000, 1)
 

Normalized defining polynomial

\( x^{15} - 270 x^{13} - 504 x^{12} + 7942 x^{11} - 91568 x^{10} - 296010 x^{9} + 2916576 x^{8} + 6356616 x^{7} - 16677328 x^{6} - 58165280 x^{5} + 4550400 x^{4} + 153260000 x^{3} + 171145600 x^{2} + 70784000 x + 10112000 \)

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:  \(39319960333779840237315944809779298304000000=2^{23}\cdot 5^{6}\cdot 37^{5}\cdot 79^{4}\cdot 233^{2}\cdot 857^{2}\cdot 1669^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $805.95$
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:  $2, 5, 37, 79, 233, 857, 1669$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{160} a^{11} - \frac{1}{16} a^{9} - \frac{1}{40} a^{8} + \frac{11}{80} a^{7} - \frac{1}{20} a^{6} - \frac{5}{16} a^{5} + \frac{7}{20} a^{4} + \frac{1}{10} a^{3} + \frac{9}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{160} a^{12} - \frac{1}{40} a^{9} + \frac{1}{80} a^{8} + \frac{1}{5} a^{7} + \frac{1}{16} a^{6} - \frac{3}{20} a^{5} - \frac{1}{40} a^{4} - \frac{1}{20} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{1600} a^{13} - \frac{1}{400} a^{10} + \frac{21}{800} a^{9} - \frac{3}{100} a^{8} - \frac{7}{160} a^{7} - \frac{23}{200} a^{6} + \frac{179}{400} a^{5} + \frac{99}{200} a^{4} + \frac{2}{5} a^{3} + \frac{3}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{1289905992697880020770998996735622188022176000} a^{14} + \frac{4757586661233045628304248222637875987917}{32247649817447000519274974918390554700554400} a^{13} - \frac{11607010891049605356120124363239289080363}{5159623970791520083083995986942488752088704} a^{12} - \frac{436564579113827309398957467379152876897463}{161238249087235002596374874591952773502772000} a^{11} + \frac{11298376585340523323449808433433832884797611}{644952996348940010385499498367811094011088000} a^{10} + \frac{695865028549957923742413535972062463223091}{40309562271808750649093718647988193375693000} a^{9} - \frac{696834279737582359448221403459579370318401}{128990599269788002077099899673562218802217600} a^{8} - \frac{8820617696515763744463061310590803813324649}{80619124543617501298187437295976386751386000} a^{7} - \frac{22053754615992808578839101206619346139271663}{161238249087235002596374874591952773502772000} a^{6} + \frac{9127428907085282261790281068664340254687617}{80619124543617501298187437295976386751386000} a^{5} + \frac{39770477172192674293639950985103617284787}{403095622718087506490937186479881933756930} a^{4} + \frac{345037500374045497960612953275078045966933}{1007739056795218766227342966199704834392325} a^{3} + \frac{516412284353674517631199353884769318121907}{1612382490872350025963748745919527735027720} a^{2} - \frac{56395239876879351130456904815278274459039}{201547811359043753245468593239940966878465} a + \frac{3727272815071915658189664853140650700}{40309562271808750649093718647988193375693}$

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:  \( 263530666050000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T97:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 70 conjugacy class representatives for [A(5)^3:2]S(3) are not computed
Character table for [A(5)^3:2]S(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 ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.4$x^{6} + 6 x^{4} + 4 x^{2} + 4 x + 14$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
2.6.10.1$x^{6} + 2 x^{5} + 2 x^{4} + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
$5$$\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}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37Data not computed
$79$79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.5.4.1$x^{5} - 79$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
233Data not computed
857Data not computed
1669Data not computed