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Properties

Label 15.11.4484222289...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{28}\cdot 5^{6}\cdot 53^{4}\cdot 71^{5}\cdot 27404129^{2}$
Root discriminant $813.04$
Ramified primes $2, 5, 53, 71, 27404129$
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![-1736704000, 10637312000, -23532339200, 21733222400, -5558538240, -2781299200, 1181988864, 112700512, -56878848, -787680, 680480, 31528, -2592, -432, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 432*x^13 - 2592*x^12 + 31528*x^11 + 680480*x^10 - 787680*x^9 - 56878848*x^8 + 112700512*x^7 + 1181988864*x^6 - 2781299200*x^5 - 5558538240*x^4 + 21733222400*x^3 - 23532339200*x^2 + 10637312000*x - 1736704000)
 
gp: K = bnfinit(x^15 - 432*x^13 - 2592*x^12 + 31528*x^11 + 680480*x^10 - 787680*x^9 - 56878848*x^8 + 112700512*x^7 + 1181988864*x^6 - 2781299200*x^5 - 5558538240*x^4 + 21733222400*x^3 - 23532339200*x^2 + 10637312000*x - 1736704000, 1)
 

Normalized defining polynomial

\( x^{15} - 432 x^{13} - 2592 x^{12} + 31528 x^{11} + 680480 x^{10} - 787680 x^{9} - 56878848 x^{8} + 112700512 x^{7} + 1181988864 x^{6} - 2781299200 x^{5} - 5558538240 x^{4} + 21733222400 x^{3} - 23532339200 x^{2} + 10637312000 x - 1736704000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44842222891195545361422304391393705984000000=2^{28}\cdot 5^{6}\cdot 53^{4}\cdot 71^{5}\cdot 27404129^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $813.04$
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:  $2, 5, 53, 71, 27404129$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{8} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7}$, $\frac{1}{32} a^{8}$, $\frac{1}{64} a^{9} - \frac{1}{2} a$, $\frac{1}{256} a^{10} + \frac{1}{32} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{5120} a^{11} + \frac{1}{640} a^{9} - \frac{1}{160} a^{8} - \frac{19}{640} a^{7} + \frac{1}{32} a^{6} - \frac{1}{32} a^{5} + \frac{1}{10} a^{4} + \frac{11}{160} a^{3} + \frac{1}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{20480} a^{12} + \frac{1}{2560} a^{10} - \frac{1}{640} a^{9} - \frac{19}{2560} a^{8} - \frac{3}{128} a^{7} + \frac{3}{128} a^{6} + \frac{1}{40} a^{5} + \frac{11}{640} a^{4} - \frac{1}{5} a^{3} - \frac{1}{16} a^{2}$, $\frac{1}{204800} a^{13} + \frac{1}{25600} a^{11} - \frac{3}{3200} a^{10} - \frac{19}{25600} a^{9} + \frac{1}{256} a^{8} + \frac{19}{1280} a^{7} - \frac{33}{800} a^{6} - \frac{309}{6400} a^{5} - \frac{19}{200} a^{4} + \frac{31}{160} a^{3} + \frac{1}{40} a^{2}$, $\frac{1}{396971284399436342120278526200354235990559948800} a^{14} - \frac{96354473424159436117596458177670091081157}{99242821099859085530069631550088558997639987200} a^{13} - \frac{207808322415161099233265322815740630967231}{12405352637482385691258703943761069874704998400} a^{12} + \frac{772976177411172587305282688273609676449027}{12405352637482385691258703943761069874704998400} a^{11} - \frac{59994669822304225571134589930569825740422347}{49621410549929542765034815775044279498819993600} a^{10} - \frac{10869849707875122605351541051007998918807559}{1550669079685298211407337992970133734338124800} a^{9} + \frac{36313438253100767217167945380066199994030133}{2481070527496477138251740788752213974940999680} a^{8} - \frac{51646560274708009400982586893025520957540251}{3101338159370596422814675985940267468676249600} a^{7} - \frac{75514298754231828309245308468153993908215917}{12405352637482385691258703943761069874704998400} a^{6} - \frac{78969568556006971162290770962161379493945839}{3101338159370596422814675985940267468676249600} a^{5} + \frac{7102703600186061577707360198709180124503067}{387667269921324552851834498242533433584531200} a^{4} - \frac{628908921501310574179690493312492095116581}{9691681748033113821295862456063335839613280} a^{3} - \frac{2115649340465242131696402060513387905558711}{38766726992132455285183449824253343358453120} a^{2} - \frac{458950292204922855610532823883705979243129}{1938336349606622764259172491212667167922656} a - \frac{41158890847139381557614416580022281671861}{242292043700827845532396561401583395990332}$

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:  $12$
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:  \( 4662175235540000000 \) (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

3.3.568.1

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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R $15$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ $15$ R ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.22.101$x^{8} + 4 x^{7} + 4 x^{6} + 6 x^{4} + 8 x^{3} + 4 x^{2} + 6$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
53Data not computed
71Data not computed
27404129Data not computed