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Properties

Label 15.15.6102017022...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{14}\cdot 13^{4}\cdot 113^{2}\cdot 139\cdot 2081603^{2}$
Root discriminant $967.61$
Ramified primes $2, 5, 7, 13, 113, 139, 2081603$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T101

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-62992384000, -330710016000, -748821964800, -954137766400, -748349521920, -371202419840, -115110115456, -21167338976, -1986852672, -30585240, 9930592, 599872, -11088, -1386, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1386*x^13 - 11088*x^12 + 599872*x^11 + 9930592*x^10 - 30585240*x^9 - 1986852672*x^8 - 21167338976*x^7 - 115110115456*x^6 - 371202419840*x^5 - 748349521920*x^4 - 954137766400*x^3 - 748821964800*x^2 - 330710016000*x - 62992384000)
 
gp: K = bnfinit(x^15 - 1386*x^13 - 11088*x^12 + 599872*x^11 + 9930592*x^10 - 30585240*x^9 - 1986852672*x^8 - 21167338976*x^7 - 115110115456*x^6 - 371202419840*x^5 - 748349521920*x^4 - 954137766400*x^3 - 748821964800*x^2 - 330710016000*x - 62992384000, 1)
 

Normalized defining polynomial

\( x^{15} - 1386 x^{13} - 11088 x^{12} + 599872 x^{11} + 9930592 x^{10} - 30585240 x^{9} - 1986852672 x^{8} - 21167338976 x^{7} - 115110115456 x^{6} - 371202419840 x^{5} - 748349521920 x^{4} - 954137766400 x^{3} - 748821964800 x^{2} - 330710016000 x - 62992384000 \)

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:  \(610201702279675842116755990922042945536000000=2^{18}\cdot 5^{6}\cdot 7^{14}\cdot 13^{4}\cdot 113^{2}\cdot 139\cdot 2081603^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $967.61$
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:  $2, 5, 7, 13, 113, 139, 2081603$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} - \frac{1}{40} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} + \frac{2}{25} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} + \frac{3}{1600} a^{7} - \frac{1}{100} a^{6} - \frac{1}{200} a^{5} + \frac{1}{50} a^{4} + \frac{57}{400} a^{3} - \frac{1}{10} a^{2} + \frac{9}{20} a$, $\frac{1}{6400} a^{10} - \frac{1}{3200} a^{8} - \frac{1}{200} a^{7} - \frac{1}{100} a^{6} - \frac{7}{800} a^{4} - \frac{3}{100} a^{3} + \frac{31}{200} a^{2} + \frac{1}{5}$, $\frac{1}{128000} a^{11} - \frac{3}{64000} a^{9} - \frac{3}{8000} a^{8} - \frac{51}{16000} a^{7} + \frac{11}{4000} a^{6} - \frac{19}{3200} a^{5} - \frac{103}{2000} a^{4} + \frac{101}{2000} a^{3} - \frac{37}{1000} a^{2} - \frac{67}{200} a + \frac{1}{50}$, $\frac{1}{16640000} a^{12} + \frac{217}{8320000} a^{10} - \frac{43}{1040000} a^{9} - \frac{37}{160000} a^{8} + \frac{2751}{520000} a^{7} - \frac{107}{16640} a^{6} + \frac{4237}{260000} a^{5} - \frac{3821}{65000} a^{4} - \frac{4057}{130000} a^{3} + \frac{5831}{26000} a^{2} - \frac{93}{500} a + \frac{8}{25}$, $\frac{1}{66560000000} a^{13} - \frac{31}{1664000000} a^{12} - \frac{94813}{33280000000} a^{11} - \frac{126039}{2080000000} a^{10} + \frac{211351}{2080000000} a^{9} + \frac{882851}{2080000000} a^{8} + \frac{1743861}{332800000} a^{7} - \frac{272717}{260000000} a^{6} + \frac{15196797}{2080000000} a^{5} + \frac{41609713}{520000000} a^{4} + \frac{4420671}{104000000} a^{3} - \frac{1933119}{13000000} a^{2} + \frac{94083}{400000} a + \frac{8429}{100000}$, $\frac{1}{110755840000000} a^{14} + \frac{9}{2129920000000} a^{13} - \frac{769773}{55377920000000} a^{12} + \frac{4250693}{13844480000000} a^{11} - \frac{5242097}{266240000000} a^{10} + \frac{451698359}{3461120000000} a^{9} + \frac{5603906557}{13844480000000} a^{8} - \frac{13626977561}{3461120000000} a^{7} - \frac{23202088291}{3461120000000} a^{6} - \frac{59953123}{54080000000} a^{5} - \frac{20961842841}{865280000000} a^{4} - \frac{624363117}{3328000000} a^{3} + \frac{562281607}{3328000000} a^{2} - \frac{88041}{1280000} a + \frac{774091}{3200000}$

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:  \( 700629720200000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T101:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed
Character table for [S(5)^3]3=S(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ $15$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
113.4.2.2$x^{4} - 113 x^{2} + 127690$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
113.5.0.1$x^{5} - x + 17$$1$$5$$0$$C_5$$[\ ]^{5}$
139Data not computed
2081603Data not computed