↑

Properties

Label 15.9.67256037220...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{21}\cdot 5^{5}\cdot 37^{5}\cdot 1741^{4}\cdot 10433^{2}\cdot 1216507^{2}$
Root discriminant $2446.33$
Ramified primes $2, 5, 37, 1741, 10433, 1216507$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![222848000, 0, -1125382400, 433160800, 1484167680, -1113803840, 7128368, 185704640, -58030848, 8741070, -1156208, 90798, -5256, 522, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 522*x^13 - 5256*x^12 + 90798*x^11 - 1156208*x^10 + 8741070*x^9 - 58030848*x^8 + 185704640*x^7 + 7128368*x^6 - 1113803840*x^5 + 1484167680*x^4 + 433160800*x^3 - 1125382400*x^2 + 222848000)
 
gp: K = bnfinit(x^15 + 522*x^13 - 5256*x^12 + 90798*x^11 - 1156208*x^10 + 8741070*x^9 - 58030848*x^8 + 185704640*x^7 + 7128368*x^6 - 1113803840*x^5 + 1484167680*x^4 + 433160800*x^3 - 1125382400*x^2 + 222848000, 1)
 

Normalized defining polynomial

\( x^{15} + 522 x^{13} - 5256 x^{12} + 90798 x^{11} - 1156208 x^{10} + 8741070 x^{9} - 58030848 x^{8} + 185704640 x^{7} + 7128368 x^{6} - 1113803840 x^{5} + 1484167680 x^{4} + 433160800 x^{3} - 1125382400 x^{2} + 222848000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-672560372209647616583874741225333544667034419200000=-\,2^{21}\cdot 5^{5}\cdot 37^{5}\cdot 1741^{4}\cdot 10433^{2}\cdot 1216507^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2446.33$
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:  $2, 5, 37, 1741, 10433, 1216507$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{80} a^{10} - \frac{1}{40} a^{9} + \frac{3}{40} a^{8} - \frac{1}{10} a^{7} + \frac{7}{40} a^{6} - \frac{9}{20} a^{5} + \frac{11}{40} a^{4} + \frac{7}{20} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{160} a^{11} + \frac{1}{80} a^{9} + \frac{1}{40} a^{8} - \frac{1}{80} a^{7} - \frac{1}{20} a^{6} - \frac{5}{16} a^{5} + \frac{9}{20} a^{4} - \frac{1}{2} a^{3} - \frac{9}{20} a^{2}$, $\frac{1}{320} a^{12} - \frac{1}{160} a^{10} + \frac{3}{80} a^{9} - \frac{13}{160} a^{8} + \frac{3}{40} a^{7} + \frac{27}{160} a^{6} - \frac{13}{40} a^{5} + \frac{19}{40} a^{4} + \frac{17}{40} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{1600} a^{13} + \frac{1}{800} a^{11} + \frac{1}{400} a^{10} + \frac{19}{800} a^{9} + \frac{9}{200} a^{8} + \frac{3}{160} a^{7} - \frac{31}{200} a^{6} + \frac{3}{10} a^{5} - \frac{69}{200} a^{4} - \frac{7}{20} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{142139181347384119469024366290795693277088527070099200} a^{14} - \frac{127811039953296267723903372926059648840480340457}{2220924708552876866703505723293682707454508235470300} a^{13} - \frac{10613366778963741590895470048825967805319243426459}{71069590673692059734512183145397846638544263535049600} a^{12} - \frac{52554567965539180054214861764088143654467729099429}{17767397668423014933628045786349461659636065883762400} a^{11} + \frac{425928633875902288143205634000376363409229259234863}{71069590673692059734512183145397846638544263535049600} a^{10} + \frac{339031092051263400175534473986456730585616033349803}{8883698834211507466814022893174730829818032941881200} a^{9} + \frac{1267895138237293718280352787862936234802216859686487}{71069590673692059734512183145397846638544263535049600} a^{8} - \frac{491086580987092509374368941286781097888515491285047}{2220924708552876866703505723293682707454508235470300} a^{7} - \frac{311330624960000775955290674477951040444385089759613}{4441849417105753733407011446587365414909016470940600} a^{6} + \frac{2956300593865037847409027748338634570419670295989203}{8883698834211507466814022893174730829818032941881200} a^{5} - \frac{320474650675389479603670552111078324923504560899997}{4441849417105753733407011446587365414909016470940600} a^{4} + \frac{97946529599334842618354154384195291516482545108207}{444184941710575373340701144658736541490901647094060} a^{3} - \frac{370330769389989707908396799757871809701191692383937}{888369883421150746681402289317473082981803294188120} a^{2} - \frac{8762738167719444822680386367953886121473803089191}{44418494171057537334070114465873654149090164709406} a + \frac{5559238382870894133161477706636893972958994475411}{22209247085528768667035057232936827074545082354703}$

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:  $11$
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:  \( 950430846326000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R $15$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.11.3$x^{6} + 2 x^{2} + 14$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37Data not computed
1741Data not computed
10433Data not computed
1216507Data not computed