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Properties

Label 15.15.2983873401...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{22}\cdot 3^{20}\cdot 5^{10}\cdot 43^{6}\cdot 5749^{2}$
Root discriminant $499.25$
Ramified primes $2, 3, 5, 43, 5749$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T84

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-45992000, 151773600, 289749600, -299868400, -557130900, -101727270, 94249160, 25190370, -4830120, -1478380, 86940, 33210, -460, -306, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 306*x^13 - 460*x^12 + 33210*x^11 + 86940*x^10 - 1478380*x^9 - 4830120*x^8 + 25190370*x^7 + 94249160*x^6 - 101727270*x^5 - 557130900*x^4 - 299868400*x^3 + 289749600*x^2 + 151773600*x - 45992000)
 
gp: K = bnfinit(x^15 - 306*x^13 - 460*x^12 + 33210*x^11 + 86940*x^10 - 1478380*x^9 - 4830120*x^8 + 25190370*x^7 + 94249160*x^6 - 101727270*x^5 - 557130900*x^4 - 299868400*x^3 + 289749600*x^2 + 151773600*x - 45992000, 1)
 

Normalized defining polynomial

\( x^{15} - 306 x^{13} - 460 x^{12} + 33210 x^{11} + 86940 x^{10} - 1478380 x^{9} - 4830120 x^{8} + 25190370 x^{7} + 94249160 x^{6} - 101727270 x^{5} - 557130900 x^{4} - 299868400 x^{3} + 289749600 x^{2} + 151773600 x - 45992000 \)

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:  \(29838734014921969199476639703040000000000=2^{22}\cdot 3^{20}\cdot 5^{10}\cdot 43^{6}\cdot 5749^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $499.25$
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:  $2, 3, 5, 43, 5749$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{10} a^{10} + \frac{2}{5} a^{8}$, $\frac{1}{20} a^{11} - \frac{3}{10} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{80} a^{12} - \frac{1}{40} a^{11} - \frac{1}{40} a^{10} - \frac{1}{10} a^{9} - \frac{7}{40} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{160} a^{13} + \frac{1}{80} a^{11} + \frac{1}{40} a^{10} + \frac{1}{80} a^{9} + \frac{19}{40} a^{8} + \frac{3}{8} a^{7} + \frac{1}{4} a^{6} + \frac{5}{16} a^{5} + \frac{1}{4} a^{4} - \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{3012947023512793154612476139267261757947537067548800} a^{14} - \frac{240615801996448933472172318317864734011216086567}{301294702351279315461247613926726175794753706754880} a^{13} - \frac{5029796911644036651274777396950030949402350620363}{1506473511756396577306238069633630878973768533774400} a^{12} + \frac{1325485062233851506429437922757948874235928450179}{75323675587819828865311903481681543948688426688720} a^{11} - \frac{279103315130155112715476519863342731050069331403}{60258940470255863092249522785345235158950741350976} a^{10} - \frac{2544026986945680440328969446569794128249860779861}{18830918896954957216327975870420385987172106672180} a^{9} - \frac{7079590729777716100697194201344258754952773262823}{30129470235127931546124761392672617579475370675488} a^{8} - \frac{5255466882842563722833527713061035165233340016617}{18830918896954957216327975870420385987172106672180} a^{7} - \frac{6074320240950027162319547617318229797784675166803}{301294702351279315461247613926726175794753706754880} a^{6} + \frac{73382666884408228716084538084285001042947538738643}{150647351175639657730623806963363087897376853377440} a^{5} - \frac{84405569927853346768266054778302559840297258503427}{301294702351279315461247613926726175794753706754880} a^{4} + \frac{1632858660629174717864884849911758131082210310651}{3766183779390991443265595174084077197434421334436} a^{3} + \frac{583693327836590548087952770836372600888074006619}{1883091889695495721632797587042038598717210667218} a^{2} - \frac{18936753552642337326658163721191964736594758250}{941545944847747860816398793521019299358605333609} a + \frac{1617271204475165874063107963128587503047020864535}{3766183779390991443265595174084077197434421334436}$

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:  \( 50410516790700000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T84:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 39 conjugacy class representatives for 1/2[S(3)^5]F(5)
Character table for 1/2[S(3)^5]F(5) is not computed

Intermediate fields

5.5.3698000.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{5}^{4}$
2.10.18.9$x^{10} - 2 x^{9} - 6 x^{8} - 6$$10$$1$$18$$(C_2^4 : C_5):C_4$$[14/5, 14/5, 14/5, 14/5]_{5}^{4}$
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.12.16.26$x^{12} + 60 x^{11} - 99 x^{10} + 9 x^{9} - 9 x^{8} - 81 x^{7} - 81 x^{6} - 27 x^{5} - 81 x^{4} - 108 x^{3} + 81 x^{2} - 81$$3$$4$$16$12T173$[2, 2, 2, 2]^{8}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.12.6.2$x^{12} - 147008443 x^{2} + 164355439274$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
5749Data not computed