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Properties

Label 15.15.8604216546...5625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{18}\cdot 7^{10}\cdot 41^{8}$
Root discriminant $182.94$
Ramified primes $5, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T30

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-774941, -6446635, -19903040, -28602215, -17230250, 1502486, 6430440, 1695760, -633040, -250100, 21517, 11395, -225, -190, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 190*x^13 - 225*x^12 + 11395*x^11 + 21517*x^10 - 250100*x^9 - 633040*x^8 + 1695760*x^7 + 6430440*x^6 + 1502486*x^5 - 17230250*x^4 - 28602215*x^3 - 19903040*x^2 - 6446635*x - 774941)
 
gp: K = bnfinit(x^15 - 190*x^13 - 225*x^12 + 11395*x^11 + 21517*x^10 - 250100*x^9 - 633040*x^8 + 1695760*x^7 + 6430440*x^6 + 1502486*x^5 - 17230250*x^4 - 28602215*x^3 - 19903040*x^2 - 6446635*x - 774941, 1)
 

Normalized defining polynomial

\( x^{15} - 190 x^{13} - 225 x^{12} + 11395 x^{11} + 21517 x^{10} - 250100 x^{9} - 633040 x^{8} + 1695760 x^{7} + 6430440 x^{6} + 1502486 x^{5} - 17230250 x^{4} - 28602215 x^{3} - 19903040 x^{2} - 6446635 x - 774941 \)

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:  \(8604216546410890678897857666015625=5^{18}\cdot 7^{10}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.94$
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:  $5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{82} a^{10} - \frac{13}{41} a^{8} + \frac{21}{82} a^{7} - \frac{3}{82} a^{6} + \frac{33}{82} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{82} a^{11} + \frac{15}{82} a^{9} - \frac{10}{41} a^{8} + \frac{19}{41} a^{7} + \frac{33}{82} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{1066} a^{12} - \frac{2}{533} a^{11} - \frac{2}{533} a^{10} + \frac{124}{533} a^{9} - \frac{227}{533} a^{8} - \frac{136}{533} a^{7} + \frac{335}{1066} a^{6} - \frac{6}{533} a^{5} - \frac{1}{2} a^{4} - \frac{7}{26} a^{3} - \frac{9}{26} a^{2} - \frac{9}{26} a - \frac{11}{26}$, $\frac{1}{1066} a^{13} + \frac{3}{533} a^{11} - \frac{1}{533} a^{10} - \frac{69}{533} a^{9} + \frac{139}{533} a^{8} - \frac{415}{1066} a^{7} - \frac{155}{533} a^{6} + \frac{225}{1066} a^{5} - \frac{7}{26} a^{4} - \frac{11}{26} a^{3} + \frac{7}{26} a^{2} + \frac{5}{26} a + \frac{4}{13}$, $\frac{1}{6806870325680204950235420731714} a^{14} + \frac{1223553806401970491368312096}{3403435162840102475117710365857} a^{13} + \frac{1246074706142987463611594599}{6806870325680204950235420731714} a^{12} + \frac{9681765715261419454162057057}{6806870325680204950235420731714} a^{11} + \frac{17365378498385840817938757667}{6806870325680204950235420731714} a^{10} - \frac{532915873222398578943616730752}{3403435162840102475117710365857} a^{9} - \frac{379789027592910139330936760144}{3403435162840102475117710365857} a^{8} - \frac{1046714189766317062688985238995}{3403435162840102475117710365857} a^{7} - \frac{414396012822165114399233045205}{3403435162840102475117710365857} a^{6} + \frac{1445105557797988748188305166941}{6806870325680204950235420731714} a^{5} - \frac{16382930463922121958630304137}{83010613727807377441895374777} a^{4} + \frac{29488563953448884457195287467}{83010613727807377441895374777} a^{3} + \frac{60277850223653532067054770491}{166021227455614754883790749554} a^{2} - \frac{862846245121798486006688678}{6385431825215952110915028829} a + \frac{52613069385775948836185760}{180066407218671100741638557}$

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:  \( 1909494312040 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T30:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 750
The 32 conjugacy class representatives for [5^3:2]3
Character table for [5^3:2]3 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 15 siblings: data not computed
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R $15$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $15$

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
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.5.4.3$x^{5} - 1476$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.4$x^{5} + 8856$$5$$1$$4$$C_5$$[\ ]_{5}$