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Properties

Label 15.7.23150300260...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{8}\cdot 5^{20}\cdot 61^{4}\cdot 89^{4}\cdot 127^{10}$
Root discriminant $3097.21$
Ramified primes $2, 5, 61, 89, 127$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group 15T39

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-655371507200, 699537508000, 500953374400, -70922827300, -75950352750, -5367093328, 2920931515, 379781060, -47194570, -8511525, 345483, 90615, -950, -475, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 475*x^13 - 950*x^12 + 90615*x^11 + 345483*x^10 - 8511525*x^9 - 47194570*x^8 + 379781060*x^7 + 2920931515*x^6 - 5367093328*x^5 - 75950352750*x^4 - 70922827300*x^3 + 500953374400*x^2 + 699537508000*x - 655371507200)
 
gp: K = bnfinit(x^15 - 475*x^13 - 950*x^12 + 90615*x^11 + 345483*x^10 - 8511525*x^9 - 47194570*x^8 + 379781060*x^7 + 2920931515*x^6 - 5367093328*x^5 - 75950352750*x^4 - 70922827300*x^3 + 500953374400*x^2 + 699537508000*x - 655371507200, 1)
 

Normalized defining polynomial

\( x^{15} - 475 x^{13} - 950 x^{12} + 90615 x^{11} + 345483 x^{10} - 8511525 x^{9} - 47194570 x^{8} + 379781060 x^{7} + 2920931515 x^{6} - 5367093328 x^{5} - 75950352750 x^{4} - 70922827300 x^{3} + 500953374400 x^{2} + 699537508000 x - 655371507200 \)

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:  \(23150300260704738550964709538721532446289062500000000=2^{8}\cdot 5^{20}\cdot 61^{4}\cdot 89^{4}\cdot 127^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3097.21$
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:  $2, 5, 61, 89, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{11} + \frac{7}{32} a^{8} - \frac{3}{32} a^{7} + \frac{1}{16} a^{6} + \frac{9}{32} a^{5} - \frac{1}{8} a^{4} + \frac{1}{16} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{160} a^{13} + \frac{3}{32} a^{9} + \frac{3}{160} a^{8} - \frac{1}{2} a^{7} - \frac{3}{32} a^{6} + \frac{3}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{20} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{429812495444398503165626996378667607759399553428739840} a^{14} + \frac{463140952029838226788773268524158403580801911691777}{429812495444398503165626996378667607759399553428739840} a^{13} + \frac{417672933064670870255876275327541174923072029895215}{42981249544439850316562699637866760775939955342873984} a^{12} - \frac{21511334357969151079073851578564105995295852108379}{5372656193054981289570337454733345096992494417859248} a^{11} + \frac{1213824425755232913832723128832954862532217131322875}{85962499088879700633125399275733521551879910685747968} a^{10} - \frac{21096556979924143238217922562238171135571298369252231}{214906247722199251582813498189333803879699776714369920} a^{9} - \frac{84003525878205211416129583935780339058784253210844259}{429812495444398503165626996378667607759399553428739840} a^{8} - \frac{28049716087782304746905524624710654610121976802718441}{85962499088879700633125399275733521551879910685747968} a^{7} - \frac{10004357435812067667759446170520746888808678854383301}{85962499088879700633125399275733521551879910685747968} a^{6} + \frac{17004247091062206183941776521476282955634739187696645}{42981249544439850316562699637866760775939955342873984} a^{5} + \frac{4181784255079837189408077466418351596180848055632121}{214906247722199251582813498189333803879699776714369920} a^{4} + \frac{32785411416360549429437968831848538916170820878270441}{107453123861099625791406749094666901939849888357184960} a^{3} + \frac{639564774789200393480204034531438763511680798474375}{2686328096527490644785168727366672548496247208929624} a^{2} - \frac{171232386658829093350680370076526736045810041651145}{2686328096527490644785168727366672548496247208929624} a - \frac{1043535446050825171216331862559314246568843680662}{335791012065936330598146090920834068562030901116203}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 4927873926650000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T39:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 28 conjugacy class representatives for [1/2.D(5)^3]3
Character table for [1/2.D(5)^3]3 is not computed

Intermediate fields

3.3.16129.1

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

Sibling fields

Degree 20 sibling: 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 $15$ R $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ $15$ $15$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $15$ $15$

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$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
$5$5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
$61$61.5.4.4$x^{5} + 488$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
61.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
89Data not computed
127Data not computed