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Properties

Label 15.1.41028151731...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 5^{10}\cdot 29^{5}\cdot 31^{13}$
Root discriminant $509.96$
Ramified primes $2, 5, 29, 31$
Class number $55$ (GRH)
Class group $[55]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1022287273, -496934421, 355869121, -169248627, 88209781, -8035207, -4879383, 3700959, -387685, -11163, -4453, 6803, -761, 23, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 23*x^13 - 761*x^12 + 6803*x^11 - 4453*x^10 - 11163*x^9 - 387685*x^8 + 3700959*x^7 - 4879383*x^6 - 8035207*x^5 + 88209781*x^4 - 169248627*x^3 + 355869121*x^2 - 496934421*x + 1022287273)
 
gp: K = bnfinit(x^15 - 5*x^14 + 23*x^13 - 761*x^12 + 6803*x^11 - 4453*x^10 - 11163*x^9 - 387685*x^8 + 3700959*x^7 - 4879383*x^6 - 8035207*x^5 + 88209781*x^4 - 169248627*x^3 + 355869121*x^2 - 496934421*x + 1022287273, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 23 x^{13} - 761 x^{12} + 6803 x^{11} - 4453 x^{10} - 11163 x^{9} - 387685 x^{8} + 3700959 x^{7} - 4879383 x^{6} - 8035207 x^{5} + 88209781 x^{4} - 169248627 x^{3} + 355869121 x^{2} - 496934421 x + 1022287273 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-41028151731920370720386831073280000000000=-\,2^{23}\cdot 5^{10}\cdot 29^{5}\cdot 31^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $509.96$
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:  $2, 5, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{16} a^{5} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{7}{16} a - \frac{3}{16}$, $\frac{1}{208644469900006263440841618427134333512326849760333422160} a^{14} + \frac{200179280273944050952795640205204044551214631537761737}{13040279368750391465052601151695895844520428110020838885} a^{13} + \frac{7549856322583353244489041300822700514815059874767913267}{208644469900006263440841618427134333512326849760333422160} a^{12} + \frac{2166173199447557154334151073743850025227426866815199759}{104322234950003131720420809213567166756163424880166711080} a^{11} + \frac{10619995052306119140253885241657589706919685508955446099}{208644469900006263440841618427134333512326849760333422160} a^{10} + \frac{1088043254658078511953868336224878737562188054529516123}{10432223495000313172042080921356716675616342488016671108} a^{9} + \frac{11139114921813650024685905398840560957545088277450536147}{208644469900006263440841618427134333512326849760333422160} a^{8} + \frac{10307788682602349712970933231525084607956835087492039861}{52161117475001565860210404606783583378081712440083355540} a^{7} + \frac{40162041851026839044015947577953011541121855021209441307}{208644469900006263440841618427134333512326849760333422160} a^{6} + \frac{8807127120314443362784904776871515883432757720470490589}{52161117475001565860210404606783583378081712440083355540} a^{5} + \frac{2268498291578434287205875385869501817978447580339579325}{41728893980001252688168323685426866702465369952066684432} a^{4} + \frac{27801812374456399843652930804123670053277472922566899723}{104322234950003131720420809213567166756163424880166711080} a^{3} + \frac{5446659299632965287461763066738430273304096875639519845}{41728893980001252688168323685426866702465369952066684432} a^{2} - \frac{2966209974656638876053780838571921592249031713783388803}{26080558737500782930105202303391791689040856220041677770} a - \frac{61583918753996428597289246954732999280546467209194724719}{208644469900006263440841618427134333512326849760333422160}$

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

Class group and class number

$C_{55}$, which has order $55$ (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:  $7$
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:  \( 21524932063325.96 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.35960.1, 5.1.1847042000.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.5.4.2$x^{5} + 217$$5$$1$$4$$C_5$$[\ ]_{5}$
31.10.9.4$x^{10} - 3647119$$10$$1$$9$$C_{10}$$[\ ]_{10}$