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Properties

Label 15.15.9192733277...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{15}\cdot 5^{6}\cdot 7^{10}\cdot 13^{4}\cdot 827^{2}\cdot 44449^{4}\cdot 28872271^{2}$
Root discriminant $11{,}593.90$
Ramified primes $2, 5, 7, 13, 827, 44449, 28872271$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T95

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-295852544000, -1553225856000, -3516947116800, -4481241502400, -3514728222720, -1743471235440, -540865750896, -99741836916, -9558128952, -224467605, 33078512, 2027042, -21528, -2691, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2691*x^13 - 21528*x^12 + 2027042*x^11 + 33078512*x^10 - 224467605*x^9 - 9558128952*x^8 - 99741836916*x^7 - 540865750896*x^6 - 1743471235440*x^5 - 3514728222720*x^4 - 4481241502400*x^3 - 3516947116800*x^2 - 1553225856000*x - 295852544000)
 
gp: K = bnfinit(x^15 - 2691*x^13 - 21528*x^12 + 2027042*x^11 + 33078512*x^10 - 224467605*x^9 - 9558128952*x^8 - 99741836916*x^7 - 540865750896*x^6 - 1743471235440*x^5 - 3514728222720*x^4 - 4481241502400*x^3 - 3516947116800*x^2 - 1553225856000*x - 295852544000, 1)
 

Normalized defining polynomial

\( x^{15} - 2691 x^{13} - 21528 x^{12} + 2027042 x^{11} + 33078512 x^{10} - 224467605 x^{9} - 9558128952 x^{8} - 99741836916 x^{7} - 540865750896 x^{6} - 1743471235440 x^{5} - 3514728222720 x^{4} - 4481241502400 x^{3} - 3516947116800 x^{2} - 1553225856000 x - 295852544000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9192733277847478572036540032446527362279872800831959552000000=2^{15}\cdot 5^{6}\cdot 7^{10}\cdot 13^{4}\cdot 827^{2}\cdot 44449^{4}\cdot 28872271^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}593.90$
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:  $2, 5, 7, 13, 827, 44449, 28872271$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{10} a^{7} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{100} a^{8} + \frac{1}{100} a^{6} + \frac{2}{25} a^{5} - \frac{13}{50} a^{4} - \frac{9}{25} a^{3} + \frac{31}{100} a^{2} + \frac{2}{5}$, $\frac{1}{800} a^{9} + \frac{21}{800} a^{7} + \frac{1}{100} a^{6} + \frac{37}{400} a^{5} + \frac{9}{50} a^{4} - \frac{149}{800} a^{3} + \frac{9}{20} a^{2} + \frac{17}{40} a - \frac{1}{2}$, $\frac{1}{1009600} a^{10} - \frac{91}{1009600} a^{8} - \frac{5139}{126200} a^{7} - \frac{7219}{504800} a^{6} - \frac{2977}{63100} a^{5} - \frac{354517}{1009600} a^{4} + \frac{12129}{126200} a^{3} + \frac{19497}{252400} a^{2} + \frac{905}{2524} a - \frac{1074}{3155}$, $\frac{1}{40384000} a^{11} - \frac{1}{2019200} a^{10} + \frac{12529}{40384000} a^{9} - \frac{24967}{10096000} a^{8} + \frac{132571}{20192000} a^{7} - \frac{40531}{5048000} a^{6} + \frac{114587}{1615360} a^{5} - \frac{2150693}{10096000} a^{4} + \frac{1975353}{5048000} a^{3} - \frac{73931}{157750} a^{2} + \frac{72419}{504800} a + \frac{61233}{126200}$, $\frac{1}{403840000} a^{12} + \frac{129}{403840000} a^{10} + \frac{31459}{50480000} a^{9} + \frac{891411}{201920000} a^{8} + \frac{60609}{3155000} a^{7} - \frac{2579993}{80768000} a^{6} - \frac{886339}{50480000} a^{5} + \frac{23597523}{50480000} a^{4} + \frac{1130951}{6310000} a^{3} - \frac{526837}{5048000} a^{2} - \frac{275301}{631000} a + \frac{7747}{63100}$, $\frac{1}{807680000000} a^{13} + \frac{19}{20192000000} a^{12} + \frac{9069}{807680000000} a^{11} + \frac{3633}{12620000000} a^{10} + \frac{240366601}{403840000000} a^{9} - \frac{108914939}{25240000000} a^{8} - \frac{896400753}{161536000000} a^{7} - \frac{3644019377}{50480000000} a^{6} - \frac{5994568409}{201920000000} a^{5} + \frac{13807562039}{50480000000} a^{4} - \frac{1503233087}{10096000000} a^{3} + \frac{50467693}{1262000000} a^{2} - \frac{31319863}{504800000} a + \frac{21419831}{126200000}$, $\frac{1}{14680391680000000} a^{14} - \frac{107}{3670097920000000} a^{13} + \frac{16466189}{14680391680000000} a^{12} - \frac{6161073}{734019584000000} a^{11} - \frac{113625527}{7340195840000000} a^{10} + \frac{1073710334747}{1835048960000000} a^{9} + \frac{9359210077259}{14680391680000000} a^{8} + \frac{54910306180697}{3670097920000000} a^{7} + \frac{30743237194219}{734019584000000} a^{6} - \frac{8105965656997}{229381120000000} a^{5} + \frac{52620032582233}{917524480000000} a^{4} - \frac{343756829871}{1835048960000} a^{3} - \frac{20587020233883}{45876224000000} a^{2} + \frac{30995550821}{1146905600000} a - \frac{248476430307}{573452800000}$

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:  $14$
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:  \( 63171683869200000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T95:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 53 conjugacy class representatives for [A(5)^3:2]3 are not computed
Character table for [A(5)^3:2]3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 sibling: 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 R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.9.4$x^{6} + 4 x^{2} + 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
827Data not computed
44449Data not computed
28872271Data not computed