↑

Properties

Label 15.15.1172475902...3125.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{8}\cdot 5^{5}\cdot 17^{4}\cdot 401^{6}\cdot 405799^{2}$
Root discriminant $402.35$
Ramified primes $3, 5, 17, 401, 405799$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T55

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46908749, -12649353, -345361931, -514816417, -284567340, -33530967, 26762204, 8799029, -511152, -498180, -17004, 12969, 720, -172, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 172*x^13 + 720*x^12 + 12969*x^11 - 17004*x^10 - 498180*x^9 - 511152*x^8 + 8799029*x^7 + 26762204*x^6 - 33530967*x^5 - 284567340*x^4 - 514816417*x^3 - 345361931*x^2 - 12649353*x + 46908749)
 
gp: K = bnfinit(x^15 - 7*x^14 - 172*x^13 + 720*x^12 + 12969*x^11 - 17004*x^10 - 498180*x^9 - 511152*x^8 + 8799029*x^7 + 26762204*x^6 - 33530967*x^5 - 284567340*x^4 - 514816417*x^3 - 345361931*x^2 - 12649353*x + 46908749, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 172 x^{13} + 720 x^{12} + 12969 x^{11} - 17004 x^{10} - 498180 x^{9} - 511152 x^{8} + 8799029 x^{7} + 26762204 x^{6} - 33530967 x^{5} - 284567340 x^{4} - 514816417 x^{3} - 345361931 x^{2} - 12649353 x + 46908749 \)

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:  \(1172475902651239286742748476173737753125=3^{8}\cdot 5^{5}\cdot 17^{4}\cdot 401^{6}\cdot 405799^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $402.35$
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:  $3, 5, 17, 401, 405799$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{51} a^{13} - \frac{4}{51} a^{12} - \frac{1}{3} a^{11} + \frac{1}{51} a^{10} - \frac{16}{51} a^{9} - \frac{4}{51} a^{8} + \frac{7}{51} a^{7} - \frac{14}{51} a^{6} + \frac{4}{51} a^{5} + \frac{25}{51} a^{4} - \frac{19}{51} a^{3} - \frac{1}{51} a^{2} + \frac{3}{17} a - \frac{16}{51}$, $\frac{1}{8545938349862671421903255361382646054925399} a^{14} - \frac{17954719621111026793182067728226821229813}{2848646116620890473967751787127548684975133} a^{13} + \frac{748545386098213136637159373917393607469841}{2848646116620890473967751787127548684975133} a^{12} - \frac{2862347311558010094479657901463090658127118}{8545938349862671421903255361382646054925399} a^{11} - \frac{1355641669374858603895133768719208819289719}{2848646116620890473967751787127548684975133} a^{10} + \frac{1082809248198491706060134272946018366777926}{8545938349862671421903255361382646054925399} a^{9} - \frac{6406004517554932698382653717496894131645}{167567418624758263174573634536914628527949} a^{8} + \frac{3937132353353174642322951283616542610983301}{8545938349862671421903255361382646054925399} a^{7} - \frac{3944981051746180685756065580490886064959078}{8545938349862671421903255361382646054925399} a^{6} + \frac{3866190994994176736651233117150004129844242}{8545938349862671421903255361382646054925399} a^{5} - \frac{1029172268241049721124162779393821519009544}{2848646116620890473967751787127548684975133} a^{4} - \frac{660775641202485790098993877448075744363696}{8545938349862671421903255361382646054925399} a^{3} + \frac{2114620876554547156114131185982279627274295}{8545938349862671421903255361382646054925399} a^{2} + \frac{1024882655028648121684547637957053929126253}{8545938349862671421903255361382646054925399} a + \frac{313294836061592360845477577563026740818124}{8545938349862671421903255361382646054925399}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 308150608616000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T55:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4860
The 48 conjugacy class representatives for [3^5:2]D(5)
Character table for [3^5:2]D(5) is not computed

Intermediate fields

5.5.160801.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.8.9$x^{6} + 6 x^{5} + 9$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$5$5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
401Data not computed
405799Data not computed