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Properties

Label 15.1.79407956764...6688.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{14}\cdot 19^{8}\cdot 433^{7}$
Root discriminant $156.07$
Ramified primes $2, 19, 433$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 15T48

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-409279744, 369095680, -183172288, 41364864, -4012116, 1741752, -1641741, 746379, -161749, 17727, -3852, 1066, -25, -13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 13*x^13 - 25*x^12 + 1066*x^11 - 3852*x^10 + 17727*x^9 - 161749*x^8 + 746379*x^7 - 1641741*x^6 + 1741752*x^5 - 4012116*x^4 + 41364864*x^3 - 183172288*x^2 + 369095680*x - 409279744)
 
gp: K = bnfinit(x^15 - 5*x^14 - 13*x^13 - 25*x^12 + 1066*x^11 - 3852*x^10 + 17727*x^9 - 161749*x^8 + 746379*x^7 - 1641741*x^6 + 1741752*x^5 - 4012116*x^4 + 41364864*x^3 - 183172288*x^2 + 369095680*x - 409279744, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 13 x^{13} - 25 x^{12} + 1066 x^{11} - 3852 x^{10} + 17727 x^{9} - 161749 x^{8} + 746379 x^{7} - 1641741 x^{6} + 1741752 x^{5} - 4012116 x^{4} + 41364864 x^{3} - 183172288 x^{2} + 369095680 x - 409279744 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-794079567645542398529000804466688=-\,2^{14}\cdot 19^{8}\cdot 433^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.07$
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:  $2, 19, 433$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{11} - \frac{1}{16} a^{10} + \frac{3}{32} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{5}{32} a^{5} + \frac{1}{8} a^{4} - \frac{7}{32} a^{3} + \frac{7}{16} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} + \frac{1}{16} a^{9} + \frac{1}{8} a^{7} + \frac{3}{32} a^{6} - \frac{3}{16} a^{5} + \frac{1}{32} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{2432} a^{13} - \frac{7}{2432} a^{12} - \frac{1}{128} a^{11} - \frac{75}{2432} a^{10} - \frac{1}{32} a^{9} + \frac{39}{608} a^{8} - \frac{521}{2432} a^{7} - \frac{275}{2432} a^{6} - \frac{1131}{2432} a^{5} + \frac{249}{2432} a^{4} - \frac{271}{1216} a^{3} + \frac{51}{152} a^{2} + \frac{65}{152} a - \frac{7}{19}$, $\frac{1}{6783610582590929305385064093323044357990389760} a^{14} - \frac{324843475477131930487778105096166307066927}{6783610582590929305385064093323044357990389760} a^{13} - \frac{23570785367151572291888697356931401785020439}{6783610582590929305385064093323044357990389760} a^{12} + \frac{24499144896796069385020765720238048043167533}{6783610582590929305385064093323044357990389760} a^{11} + \frac{832150794601562691178737019503857054390013}{169590264564773232634626602333076108949759744} a^{10} + \frac{47875507986759152473059710490777880274835817}{1695902645647732326346266023330761089497597440} a^{9} - \frac{339295244150955158236235669173399188163677609}{6783610582590929305385064093323044357990389760} a^{8} - \frac{1442279736082358449684872063929060782850664731}{6783610582590929305385064093323044357990389760} a^{7} - \frac{459683725198161557406976525218372445941340999}{6783610582590929305385064093323044357990389760} a^{6} + \frac{678905550978151149604324583024712590712670617}{6783610582590929305385064093323044357990389760} a^{5} + \frac{155303538843777160447720151249729052709681519}{3391805291295464652692532046661522178995194880} a^{4} + \frac{26047301849661413987046147581901342778855427}{105993915352983270396641626458172568093599840} a^{3} - \frac{3406907474145044869559347212229204365873221}{13249239419122908799580203307271571011699980} a^{2} - \frac{1166530851749563252098600092695973868399651}{105993915352983270396641626458172568093599840} a - \frac{12501767161551654410686049314970988178138809}{52996957676491635198320813229086284046799920}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $7$
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:  \( 463572992103 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T48:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 26 conjugacy class representatives for 1/2[D(5)^3]S(3)
Character table for 1/2[D(5)^3]S(3) is not computed

Intermediate fields

3.1.1732.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.5.0.1}{5} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{5}$ $15$ $15$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{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}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
433Data not computed