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Properties

Label 15.1.19194711673...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 19^{13}\cdot 31^{5}$
Root discriminant $484.78$
Ramified primes $2, 3, 5, 19, 31$
Class number $25$ (GRH)
Class group $[5, 5]$ (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![-28629151, -21240983, -38132480, -48976404, -49323325, -1834549, -6668441, -167121, 126897, -6733, 16969, 1585, 48, 80, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 80*x^13 + 48*x^12 + 1585*x^11 + 16969*x^10 - 6733*x^9 + 126897*x^8 - 167121*x^7 - 6668441*x^6 - 1834549*x^5 - 49323325*x^4 - 48976404*x^3 - 38132480*x^2 - 21240983*x - 28629151)
 
gp: K = bnfinit(x^15 - x^14 + 80*x^13 + 48*x^12 + 1585*x^11 + 16969*x^10 - 6733*x^9 + 126897*x^8 - 167121*x^7 - 6668441*x^6 - 1834549*x^5 - 49323325*x^4 - 48976404*x^3 - 38132480*x^2 - 21240983*x - 28629151, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} + 80 x^{13} + 48 x^{12} + 1585 x^{11} + 16969 x^{10} - 6733 x^{9} + 126897 x^{8} - 167121 x^{7} - 6668441 x^{6} - 1834549 x^{5} - 49323325 x^{4} - 48976404 x^{3} - 38132480 x^{2} - 21240983 x - 28629151 \)

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:  \(-19194711673970613594516583376070000000000=-\,2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 19^{13}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $484.78$
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:  $2, 3, 5, 19, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{62} a^{8} + \frac{11}{62} a^{7} - \frac{5}{62} a^{6} - \frac{6}{31} a^{5} - \frac{8}{31} a^{4} + \frac{3}{31} a^{3} - \frac{27}{62} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{62} a^{9} - \frac{1}{31} a^{7} - \frac{19}{62} a^{6} - \frac{4}{31} a^{5} - \frac{2}{31} a^{4} - \frac{1}{2} a^{3} + \frac{9}{31} a^{2} - \frac{1}{2}$, $\frac{1}{62} a^{10} + \frac{3}{62} a^{7} - \frac{9}{31} a^{6} - \frac{14}{31} a^{5} - \frac{1}{62} a^{4} + \frac{15}{31} a^{3} + \frac{4}{31} a^{2} - \frac{1}{2} a$, $\frac{1}{1922} a^{11} - \frac{1}{1922} a^{10} - \frac{13}{1922} a^{9} - \frac{7}{961} a^{8} + \frac{64}{961} a^{7} - \frac{87}{961} a^{6} + \frac{521}{1922} a^{5} - \frac{513}{1922} a^{4} - \frac{9}{62} a^{3} - \frac{1}{31} a^{2}$, $\frac{1}{27467302} a^{12} - \frac{2791}{27467302} a^{11} + \frac{14207}{13733651} a^{10} - \frac{153557}{27467302} a^{9} - \frac{106915}{27467302} a^{8} - \frac{1241693}{27467302} a^{7} - \frac{4556603}{27467302} a^{6} - \frac{2887116}{13733651} a^{5} + \frac{77995}{886042} a^{4} + \frac{102487}{443021} a^{3} + \frac{6587}{14291} a^{2} + \frac{5433}{14291} a + \frac{31}{922}$, $\frac{1}{137336510} a^{13} - \frac{1}{68668255} a^{12} + \frac{2164}{68668255} a^{11} - \frac{879347}{137336510} a^{10} + \frac{103858}{13733651} a^{9} + \frac{198776}{68668255} a^{8} + \frac{1364922}{13733651} a^{7} + \frac{15165427}{68668255} a^{6} + \frac{11978205}{27467302} a^{5} + \frac{342274}{2215105} a^{4} + \frac{1055393}{4430210} a^{3} - \frac{6653}{28582} a^{2} - \frac{2488}{71455} a - \frac{209}{4610}$, $\frac{1}{1636874278175662020192948062135660} a^{14} + \frac{181187099260632819495164}{409218569543915505048237015533915} a^{13} - \frac{5600701540581175661289317}{409218569543915505048237015533915} a^{12} + \frac{170501040023763095723770591171}{818437139087831010096474031067830} a^{11} - \frac{12743166741671939591705887149151}{1636874278175662020192948062135660} a^{10} - \frac{1246385550088937985092414029469}{818437139087831010096474031067830} a^{9} - \frac{8975339222227230795421979578299}{1636874278175662020192948062135660} a^{8} - \frac{44129014223280228081216264135939}{409218569543915505048237015533915} a^{7} - \frac{5943170046192281864637288073023}{52802396070182645812675743939860} a^{6} - \frac{124303027338543171208895825171}{26401198035091322906337871969930} a^{5} - \frac{479371276668280173690411328093}{1703303099038149864925023998060} a^{4} + \frac{93902086512487827325109454847}{851651549519074932462511999030} a^{3} + \frac{10052384022535626978901021967}{27472630629647578466532645130} a^{2} + \frac{5632846137855320106658464497}{13736315314823789233266322565} a - \frac{621609885686252882842805447}{1772427782557908288163396460}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 42285044175841.24 \) (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

3.1.35340.1, 5.1.1319500125.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{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}$
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.10.9.2$x^{10} + 76$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$