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Properties

Label 15.7.28561623804...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 13\cdot 29^{4}\cdot 41^{4}\cdot 211^{2}\cdot 659^{2}\cdot 701^{2}$
Root discriminant $1457.87$
Ramified primes $2, 5, 7, 13, 29, 41, 211, 659, 701$
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![-4870144000, -25568256000, -57893836800, -73767462400, -57857310720, -28692461440, -8877159296, -1605430816, -132034752, 5330520, 2057312, 118592, -5328, -666, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 666*x^13 - 5328*x^12 + 118592*x^11 + 2057312*x^10 + 5330520*x^9 - 132034752*x^8 - 1605430816*x^7 - 8877159296*x^6 - 28692461440*x^5 - 57857310720*x^4 - 73767462400*x^3 - 57893836800*x^2 - 25568256000*x - 4870144000)
 
gp: K = bnfinit(x^15 - 666*x^13 - 5328*x^12 + 118592*x^11 + 2057312*x^10 + 5330520*x^9 - 132034752*x^8 - 1605430816*x^7 - 8877159296*x^6 - 28692461440*x^5 - 57857310720*x^4 - 73767462400*x^3 - 57893836800*x^2 - 25568256000*x - 4870144000, 1)
 

Normalized defining polynomial

\( x^{15} - 666 x^{13} - 5328 x^{12} + 118592 x^{11} + 2057312 x^{10} + 5330520 x^{9} - 132034752 x^{8} - 1605430816 x^{7} - 8877159296 x^{6} - 28692461440 x^{5} - 57857310720 x^{4} - 73767462400 x^{3} - 57893836800 x^{2} - 25568256000 x - 4870144000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(285616238046894465829329548900066635436032000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 13\cdot 29^{4}\cdot 41^{4}\cdot 211^{2}\cdot 659^{2}\cdot 701^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1457.87$
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:  $2, 5, 7, 13, 29, 41, 211, 659, 701$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} - \frac{1}{40} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} + \frac{2}{25} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} + \frac{3}{1600} a^{7} - \frac{1}{100} a^{6} - \frac{1}{200} a^{5} + \frac{1}{50} a^{4} + \frac{7}{400} a^{3} - \frac{1}{10} a^{2} - \frac{1}{20} a$, $\frac{1}{6400} a^{10} - \frac{1}{3200} a^{8} - \frac{1}{200} a^{7} - \frac{1}{100} a^{6} - \frac{57}{800} a^{4} - \frac{3}{100} a^{3} - \frac{19}{200} a^{2} + \frac{1}{5}$, $\frac{1}{128000} a^{11} - \frac{3}{64000} a^{9} - \frac{3}{8000} a^{8} + \frac{49}{16000} a^{7} + \frac{11}{4000} a^{6} - \frac{29}{3200} a^{5} - \frac{53}{2000} a^{4} - \frac{499}{2000} a^{3} - \frac{187}{1000} a^{2} + \frac{83}{200} a + \frac{1}{50}$, $\frac{1}{1280000} a^{12} - \frac{23}{640000} a^{10} - \frac{1}{10000} a^{9} + \frac{59}{160000} a^{8} + \frac{9}{10000} a^{7} - \frac{221}{32000} a^{6} - \frac{233}{20000} a^{5} - \frac{447}{10000} a^{4} - \frac{681}{5000} a^{3} + \frac{281}{2000} a^{2} - \frac{36}{125} a - \frac{6}{25}$, $\frac{1}{5120000000} a^{13} - \frac{31}{128000000} a^{12} + \frac{5547}{2560000000} a^{11} - \frac{3759}{160000000} a^{10} + \frac{211}{160000000} a^{9} + \frac{35211}{160000000} a^{8} - \frac{477841}{128000000} a^{7} + \frac{221903}{20000000} a^{6} + \frac{2091827}{160000000} a^{5} - \frac{1566217}{40000000} a^{4} + \frac{194361}{8000000} a^{3} + \frac{217471}{1000000} a^{2} - \frac{78111}{400000} a - \frac{21393}{100000}$, $\frac{1}{655360000000} a^{14} - \frac{11}{163840000000} a^{13} + \frac{60027}{327680000000} a^{12} - \frac{256483}{81920000000} a^{11} + \frac{165947}{20480000000} a^{10} + \frac{1869567}{20480000000} a^{9} + \frac{2788219}{81920000000} a^{8} + \frac{62714929}{20480000000} a^{7} - \frac{200930269}{20480000000} a^{6} - \frac{6078493}{640000000} a^{5} + \frac{229052273}{5120000000} a^{4} + \frac{31683681}{256000000} a^{3} + \frac{3298077}{256000000} a^{2} + \frac{1490509}{6400000} a + \frac{45493}{3200000}$

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:  $10$
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:  \( 2448997535060000000 \) (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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R $15$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ R ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]^{3}$
2.6.9.4$x^{6} + 4 x^{2} + 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.5.4.3$x^{5} - 1476$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.0.1$x^{5} - x + 7$$1$$5$$0$$C_5$$[\ ]^{5}$
211Data not computed
659Data not computed
701Data not computed