↑

Properties

Label 15.3.30898205218...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{15}\cdot 5^{17}\cdot 19^{12}\cdot 89^{5}$
Root discriminant $583.44$
Ramified primes $2, 5, 19, 89$
Class number $50$ (GRH)
Class group $[5, 10]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27711922, -17766115, 43193990, -24629270, 2985455, 4494657, -2045090, -241460, 294580, -20720, -16564, 2265, 460, -80, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 16564*x^10 - 20720*x^9 + 294580*x^8 - 241460*x^7 - 2045090*x^6 + 4494657*x^5 + 2985455*x^4 - 24629270*x^3 + 43193990*x^2 - 17766115*x - 27711922)
 
gp: K = bnfinit(x^15 - 5*x^14 - 80*x^13 + 460*x^12 + 2265*x^11 - 16564*x^10 - 20720*x^9 + 294580*x^8 - 241460*x^7 - 2045090*x^6 + 4494657*x^5 + 2985455*x^4 - 24629270*x^3 + 43193990*x^2 - 17766115*x - 27711922, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 80 x^{13} + 460 x^{12} + 2265 x^{11} - 16564 x^{10} - 20720 x^{9} + 294580 x^{8} - 241460 x^{7} - 2045090 x^{6} + 4494657 x^{5} + 2985455 x^{4} - 24629270 x^{3} + 43193990 x^{2} - 17766115 x - 27711922 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(308982052185601664705132225000000000000000=2^{15}\cdot 5^{17}\cdot 19^{12}\cdot 89^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $583.44$
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:  $2, 5, 19, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{3}$, $\frac{1}{25} a^{9} - \frac{2}{25} a^{8} - \frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{1}{25} a^{5} + \frac{2}{25} a^{4} - \frac{4}{25} a^{3} - \frac{2}{25} a^{2} + \frac{4}{25} a + \frac{2}{25}$, $\frac{1}{25} a^{10} - \frac{1}{25} a^{5} - \frac{6}{25}$, $\frac{1}{25} a^{11} - \frac{1}{25} a^{6} - \frac{6}{25} a$, $\frac{1}{125} a^{12} - \frac{1}{125} a^{11} - \frac{1}{125} a^{10} + \frac{4}{125} a^{7} - \frac{4}{125} a^{6} - \frac{4}{125} a^{5} + \frac{4}{125} a^{2} - \frac{4}{125} a - \frac{4}{125}$, $\frac{1}{125} a^{13} - \frac{2}{125} a^{11} - \frac{1}{125} a^{10} + \frac{4}{125} a^{8} - \frac{8}{125} a^{6} - \frac{4}{125} a^{5} + \frac{4}{125} a^{3} - \frac{8}{125} a - \frac{4}{125}$, $\frac{1}{103581184964043938014407378956270125} a^{14} - \frac{154341823670123580223071132276831}{103581184964043938014407378956270125} a^{13} - \frac{2674997215455490404906392334486}{103581184964043938014407378956270125} a^{12} - \frac{12563247480929633955180373299752}{828649479712351504115259031650161} a^{11} - \frac{216372710468905104861219076801947}{20716236992808787602881475791254025} a^{10} + \frac{1684867553082394001742309609159654}{103581184964043938014407378956270125} a^{9} - \frac{2092827440386897249002138400221324}{103581184964043938014407378956270125} a^{8} - \frac{7436603810207117454018237143604744}{103581184964043938014407378956270125} a^{7} + \frac{11206846101063339875715900827156}{4143247398561757520576295158250805} a^{6} - \frac{1824601408670088926477911414212088}{20716236992808787602881475791254025} a^{5} + \frac{704570243409504002618193027362679}{103581184964043938014407378956270125} a^{4} - \frac{9168175911663240392336956656335324}{103581184964043938014407378956270125} a^{3} - \frac{33517272515004321651249968641449419}{103581184964043938014407378956270125} a^{2} + \frac{673892586656711215034031754048012}{4143247398561757520576295158250805} a - \frac{8930715802005062563608168803725113}{20716236992808787602881475791254025}$

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

Class group and class number

$C_{5}\times C_{10}$, which has order $50$ (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:  $8$
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:  \( 58178720157067.875 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.3.17800.1, 5.1.407253125.2

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

Sibling fields

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$