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Properties

Label 15.9.21930462674...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{10}\cdot 5^{5}\cdot 13^{3}\cdot 19^{2}\cdot 31^{2}\cdot 347^{3}\cdot 4639^{2}$
Root discriminant $105.37$
Ramified primes $2, 5, 13, 19, 31, 347, 4639$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T83

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1265140391, 110428690, 376105913, 2783240, -33152761, -6248483, 926663, 482688, 35528, -4807, -6311, 528, 277, -62, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 62*x^13 + 277*x^12 + 528*x^11 - 6311*x^10 - 4807*x^9 + 35528*x^8 + 482688*x^7 + 926663*x^6 - 6248483*x^5 - 33152761*x^4 + 2783240*x^3 + 376105913*x^2 + 110428690*x - 1265140391)
 
gp: K = bnfinit(x^15 - 2*x^14 - 62*x^13 + 277*x^12 + 528*x^11 - 6311*x^10 - 4807*x^9 + 35528*x^8 + 482688*x^7 + 926663*x^6 - 6248483*x^5 - 33152761*x^4 + 2783240*x^3 + 376105913*x^2 + 110428690*x - 1265140391, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 62 x^{13} + 277 x^{12} + 528 x^{11} - 6311 x^{10} - 4807 x^{9} + 35528 x^{8} + 482688 x^{7} + 926663 x^{6} - 6248483 x^{5} - 33152761 x^{4} + 2783240 x^{3} + 376105913 x^{2} + 110428690 x - 1265140391 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2193046267403549887000547200000=-\,2^{10}\cdot 5^{5}\cdot 13^{3}\cdot 19^{2}\cdot 31^{2}\cdot 347^{3}\cdot 4639^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.37$
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:  $2, 5, 13, 19, 31, 347, 4639$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{531648249366447696373321770723417021718336463793910369001} a^{14} + \frac{225737096630608562492570356735347435673982599360846432005}{531648249366447696373321770723417021718336463793910369001} a^{13} + \frac{90406572675694630776028818178058630807989682593245079425}{531648249366447696373321770723417021718336463793910369001} a^{12} - \frac{205085231188792322445998879528361046339421891643938228965}{531648249366447696373321770723417021718336463793910369001} a^{11} + \frac{125387790845890657396774436756368586723927477254586457878}{531648249366447696373321770723417021718336463793910369001} a^{10} - \frac{153810108466004899626356431902762037184666146338751620542}{531648249366447696373321770723417021718336463793910369001} a^{9} - \frac{18295512641200588091849410432398167410709456753535422610}{531648249366447696373321770723417021718336463793910369001} a^{8} - \frac{211903010352053277287980098791808571032581421192832187486}{531648249366447696373321770723417021718336463793910369001} a^{7} - \frac{204933514454840489427989331327204860282172017472616528415}{531648249366447696373321770723417021718336463793910369001} a^{6} - \frac{48981129009695009993710097441944771162490021227920154588}{531648249366447696373321770723417021718336463793910369001} a^{5} + \frac{136097710761227890347876987696855853817008634548685286883}{531648249366447696373321770723417021718336463793910369001} a^{4} - \frac{125677372669321976209378912318768278917168018805500960753}{531648249366447696373321770723417021718336463793910369001} a^{3} + \frac{170095712374102961665742553266319230170676108548330873047}{531648249366447696373321770723417021718336463793910369001} a^{2} + \frac{119507130980896514875581728457923952468512112531781621763}{531648249366447696373321770723417021718336463793910369001} a - \frac{253032079596974125946216529977690054060829194395685422907}{531648249366447696373321770723417021718336463793910369001}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $11$
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:  \( 1679365862.11 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T83:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 58320
The 72 conjugacy class representatives for [3^5:2]S(5) are not computed
Character table for [3^5:2]S(5) is not computed

Intermediate fields

5.3.4511.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
2Data not computed
$5$5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
31Data not computed
347Data not computed
4639Data not computed