↑

Properties

Label 15.1.53435954791...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 3^{12}\cdot 5^{17}\cdot 7^{13}\cdot 139^{5}$
Root discriminant $959.08$
Ramified primes $2, 3, 5, 7, 139$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28154663716, 43463765000, -1867556300, 3376407000, 472707250, 207170233, -38235000, 18512525, -2346000, 277250, -52732, 6250, -550, 125, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 125*x^13 - 550*x^12 + 6250*x^11 - 52732*x^10 + 277250*x^9 - 2346000*x^8 + 18512525*x^7 - 38235000*x^6 + 207170233*x^5 + 472707250*x^4 + 3376407000*x^3 - 1867556300*x^2 + 43463765000*x + 28154663716)
 
gp: K = bnfinit(x^15 + 125*x^13 - 550*x^12 + 6250*x^11 - 52732*x^10 + 277250*x^9 - 2346000*x^8 + 18512525*x^7 - 38235000*x^6 + 207170233*x^5 + 472707250*x^4 + 3376407000*x^3 - 1867556300*x^2 + 43463765000*x + 28154663716, 1)
 

Normalized defining polynomial

\( x^{15} + 125 x^{13} - 550 x^{12} + 6250 x^{11} - 52732 x^{10} + 277250 x^{9} - 2346000 x^{8} + 18512525 x^{7} - 38235000 x^{6} + 207170233 x^{5} + 472707250 x^{4} + 3376407000 x^{3} - 1867556300 x^{2} + 43463765000 x + 28154663716 \)

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:  \(-534359547919362296589172482600000000000000000=-\,2^{18}\cdot 3^{12}\cdot 5^{17}\cdot 7^{13}\cdot 139^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $959.08$
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:  $2, 3, 5, 7, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{30} a^{6} - \frac{1}{10} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{2}{15} a - \frac{4}{15}$, $\frac{1}{30} a^{7} + \frac{1}{15} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{2}{15} a^{2} + \frac{1}{3} a + \frac{2}{5}$, $\frac{1}{30} a^{8} + \frac{1}{15} a^{5} - \frac{1}{2} a^{4} - \frac{7}{15} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{4}{15}$, $\frac{1}{150} a^{9} - \frac{1}{150} a^{8} + \frac{1}{150} a^{7} - \frac{1}{150} a^{6} + \frac{1}{25} a^{5} + \frac{1}{25} a^{4} + \frac{22}{75} a^{3} + \frac{28}{75} a^{2} - \frac{1}{25} a - \frac{32}{75}$, $\frac{1}{4200} a^{10} - \frac{3}{1400} a^{9} - \frac{1}{75} a^{8} + \frac{13}{2100} a^{7} + \frac{23}{1400} a^{6} - \frac{99}{1400} a^{5} - \frac{377}{2100} a^{4} + \frac{86}{175} a^{3} - \frac{8}{25} a^{2} + \frac{41}{1050} a + \frac{433}{1050}$, $\frac{1}{4200} a^{11} + \frac{1}{1400} a^{9} - \frac{29}{2100} a^{8} + \frac{23}{4200} a^{7} + \frac{11}{1050} a^{6} - \frac{69}{1400} a^{5} - \frac{191}{2100} a^{4} + \frac{19}{525} a^{3} - \frac{253}{1050} a^{2} + \frac{191}{525} a - \frac{23}{1050}$, $\frac{1}{63000} a^{12} - \frac{1}{21000} a^{11} + \frac{1}{63000} a^{10} + \frac{1}{360} a^{9} + \frac{1}{280} a^{8} + \frac{47}{3000} a^{7} + \frac{139}{63000} a^{6} - \frac{203}{3000} a^{5} - \frac{379}{1260} a^{4} + \frac{191}{630} a^{3} - \frac{57}{250} a^{2} + \frac{6673}{15750} a - \frac{6341}{15750}$, $\frac{1}{126000} a^{13} - \frac{1}{15750} a^{11} - \frac{1}{63000} a^{10} - \frac{1}{840} a^{9} + \frac{277}{21000} a^{8} + \frac{1}{1260} a^{7} - \frac{191}{21000} a^{6} + \frac{343}{18000} a^{5} + \frac{817}{2520} a^{4} + \frac{1289}{5250} a^{3} - \frac{146}{315} a^{2} + \frac{3989}{15750} a + \frac{19}{10500}$, $\frac{1}{1342870387710694744401081568901882180803842000} a^{14} + \frac{3425714817528088894333203667191709288511}{1342870387710694744401081568901882180803842000} a^{13} + \frac{29455551728015979866512179270727170493}{15986552234651127909536685344070025961950500} a^{12} + \frac{3195138131995097478886791623755519732193}{44762346257023158146702718963396072693461400} a^{11} - \frac{3423081223356285640600769256573321216409}{167858798463836843050135196112735272600480250} a^{10} - \frac{756458976085268041392421177899977800568617}{335717596927673686100270392225470545200960500} a^{9} + \frac{1885202437273636612803820662172389703486781}{671435193855347372200540784450941090401921000} a^{8} - \frac{5466110852145556846588612205292720158901743}{671435193855347372200540784450941090401921000} a^{7} - \frac{484276573982472430467856278971245884076637}{29841564171348772097801812642264048462307600} a^{6} + \frac{17787388447059270660190197887945087303363383}{191838626815813534914440224128840311543406000} a^{5} + \frac{33980314647038434061361801775722524143624967}{74603910428371930244504531605660121155769000} a^{4} - \frac{49332401038145221260925067712837346458354503}{167858798463836843050135196112735272600480250} a^{3} - \frac{15923255389406956035178030617183010643379427}{55952932821278947683378398704245090866826750} a^{2} - \frac{6426035360974693576677502123378141693957309}{67143519385534737220054078445094109040192100} a - \frac{12367810967470303927147888310702397794597593}{335717596927673686100270392225470545200960500}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  \( 10320283429411996 \) (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

3.1.97300.3, 5.1.9724050000.9

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 R ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{5}^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$7$7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$