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Properties

Label 15.3.32879674255...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 31^{13}\cdot 61^{5}$
Root discriminant $928.53$
Ramified primes $2, 3, 5, 31, 61$
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![-844596301, 512296117, -767195780, -911101734, 812573375, -34006219, -127033781, 2668689, 6419037, -27463, -126671, 3775, 1038, -130, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 130*x^13 + 1038*x^12 + 3775*x^11 - 126671*x^10 - 27463*x^9 + 6419037*x^8 + 2668689*x^7 - 127033781*x^6 - 34006219*x^5 + 812573375*x^4 - 911101734*x^3 - 767195780*x^2 + 512296117*x - 844596301)
 
gp: K = bnfinit(x^15 - x^14 - 130*x^13 + 1038*x^12 + 3775*x^11 - 126671*x^10 - 27463*x^9 + 6419037*x^8 + 2668689*x^7 - 127033781*x^6 - 34006219*x^5 + 812573375*x^4 - 911101734*x^3 - 767195780*x^2 + 512296117*x - 844596301, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 130 x^{13} + 1038 x^{12} + 3775 x^{11} - 126671 x^{10} - 27463 x^{9} + 6419037 x^{8} + 2668689 x^{7} - 127033781 x^{6} - 34006219 x^{5} + 812573375 x^{4} - 911101734 x^{3} - 767195780 x^{2} + 512296117 x - 844596301 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(328796742550936048048723892663067930000000000=2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 31^{13}\cdot 61^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $928.53$
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:  $2, 3, 5, 31, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{122} a^{8} - \frac{19}{122} a^{7} + \frac{29}{122} a^{6} + \frac{14}{61} a^{5} + \frac{19}{61} a^{4} - \frac{24}{61} a^{3} - \frac{3}{122} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{122} a^{9} - \frac{27}{122} a^{7} - \frac{31}{122} a^{6} + \frac{21}{122} a^{5} + \frac{3}{122} a^{4} - \frac{57}{122} a^{2}$, $\frac{1}{122} a^{10} + \frac{5}{122} a^{7} - \frac{25}{61} a^{6} - \frac{17}{61} a^{5} - \frac{11}{122} a^{4} + \frac{25}{61} a^{3} - \frac{10}{61} a^{2} - \frac{1}{2} a$, $\frac{1}{7442} a^{11} - \frac{1}{7442} a^{10} - \frac{4}{3721} a^{9} + \frac{1}{7442} a^{8} + \frac{1579}{7442} a^{7} + \frac{1599}{3721} a^{6} - \frac{1843}{7442} a^{5} + \frac{2203}{7442} a^{4} - \frac{13}{61} a^{3} + \frac{17}{122} a^{2} - \frac{1}{2} a$, $\frac{1}{453962} a^{12} - \frac{1}{453962} a^{11} + \frac{907}{453962} a^{10} + \frac{1}{453962} a^{9} - \frac{400}{226981} a^{8} - \frac{99587}{453962} a^{7} + \frac{53338}{226981} a^{6} + \frac{80615}{226981} a^{5} - \frac{196}{3721} a^{4} - \frac{358}{3721} a^{3} - \frac{35}{122} a^{2} - \frac{19}{61} a$, $\frac{1}{2269810} a^{13} - \frac{1}{1134905} a^{12} - \frac{7}{2269810} a^{11} - \frac{1856}{1134905} a^{10} - \frac{1673}{453962} a^{9} + \frac{8207}{2269810} a^{8} + \frac{99837}{453962} a^{7} + \frac{905199}{2269810} a^{6} + \frac{93189}{226981} a^{5} + \frac{303}{37210} a^{4} + \frac{2554}{18605} a^{3} - \frac{28}{61} a^{2} + \frac{99}{610} a - \frac{2}{5}$, $\frac{1}{12066717787445877294395010479271458987348920} a^{14} - \frac{257758314749695275303991518402547433}{1508339723430734661799376309908932373418615} a^{13} - \frac{3626898542141193080783113000963713689}{6033358893722938647197505239635729493674460} a^{12} - \frac{63712669060539396944778978476487207347}{3016679446861469323598752619817864746837230} a^{11} - \frac{11768693777134218624023149715351890283881}{12066717787445877294395010479271458987348920} a^{10} + \frac{4038191274522884034369200246270676672343}{3016679446861469323598752619817864746837230} a^{9} - \frac{14941104487357380800482439819153556074499}{12066717787445877294395010479271458987348920} a^{8} - \frac{1261070865819967338906367998512455818521603}{6033358893722938647197505239635729493674460} a^{7} - \frac{43430603332037623984123095250920000003193}{197815045695834054006475581627400967005720} a^{6} + \frac{4300708571677911668120417008729263263669}{98907522847917027003237790813700483502860} a^{5} - \frac{768256525190617699774157609841200765533}{3242869601571050065679927567662310934520} a^{4} + \frac{4446504613560378941752001560580117943}{18218368548151966661123188582372533340} a^{3} - \frac{2163165022188728878452608163811129752}{6645224593383299314917884359963751915} a^{2} - \frac{5590012312846438609033214648181169333}{13290449186766598629835768719927503830} a - \frac{218783994786361479504057578150365727}{871504864706006467530214342290328120}$

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:  $8$
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:  \( 5767420989072087.0 \) (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

3.3.113460.1, 5.1.9350650125.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$[\ ]_{5}^{4}$
3.10.9.2$x^{10} + 3$$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}$
$31$31.5.4.3$x^{5} - 1519$$5$$1$$4$$C_5$$[\ ]_{5}$
31.10.9.2$x^{10} - 1519$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$