↑

Properties

Label 15.1.77667356418...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 5^{17}\cdot 7^{13}\cdot 29^{13}$
Root discriminant $983.29$
Ramified primes $2, 5, 7, 29$
Class number $120$ (GRH)
Class group $[2, 2, 30]$ (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![4449241879687, 2061802323020, 184194250760, -57993380220, -8431068230, 2772478367, 473102930, -13206720, -3083670, -183910, 30349, 2070, -850, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 10*x^13 - 850*x^12 + 2070*x^11 + 30349*x^10 - 183910*x^9 - 3083670*x^8 - 13206720*x^7 + 473102930*x^6 + 2772478367*x^5 - 8431068230*x^4 - 57993380220*x^3 + 184194250760*x^2 + 2061802323020*x + 4449241879687)
 
gp: K = bnfinit(x^15 + 10*x^13 - 850*x^12 + 2070*x^11 + 30349*x^10 - 183910*x^9 - 3083670*x^8 - 13206720*x^7 + 473102930*x^6 + 2772478367*x^5 - 8431068230*x^4 - 57993380220*x^3 + 184194250760*x^2 + 2061802323020*x + 4449241879687, 1)
 

Normalized defining polynomial

\( x^{15} + 10 x^{13} - 850 x^{12} + 2070 x^{11} + 30349 x^{10} - 183910 x^{9} - 3083670 x^{8} - 13206720 x^{7} + 473102930 x^{6} + 2772478367 x^{5} - 8431068230 x^{4} - 57993380220 x^{3} + 184194250760 x^{2} + 2061802323020 x + 4449241879687 \)

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:  \(-776673564181413299414214047596093750000000000=-\,2^{10}\cdot 5^{17}\cdot 7^{13}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $983.29$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{8} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{9} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{10} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{77} a^{11} + \frac{4}{77} a^{10} - \frac{1}{77} a^{9} - \frac{3}{77} a^{8} + \frac{3}{77} a^{7} - \frac{8}{77} a^{6} + \frac{12}{77} a^{5} - \frac{34}{77} a^{4} - \frac{24}{77} a^{3} - \frac{20}{77} a^{2} + \frac{4}{77} a + \frac{3}{7}$, $\frac{1}{539} a^{12} + \frac{1}{539} a^{11} + \frac{20}{539} a^{10} + \frac{1}{49} a^{9} + \frac{34}{539} a^{8} + \frac{5}{539} a^{7} + \frac{13}{77} a^{6} + \frac{205}{539} a^{5} + \frac{177}{539} a^{4} - \frac{135}{539} a^{3} - \frac{13}{539} a^{2} - \frac{23}{539} a + \frac{13}{49}$, $\frac{1}{5929} a^{13} + \frac{1}{5929} a^{12} - \frac{2}{539} a^{11} - \frac{3}{5929} a^{10} - \frac{309}{5929} a^{9} - \frac{23}{5929} a^{8} + \frac{39}{847} a^{7} + \frac{772}{5929} a^{6} + \frac{1752}{5929} a^{5} + \frac{1524}{5929} a^{4} - \frac{1546}{5929} a^{3} - \frac{2263}{5929} a^{2} + \frac{2747}{5929} a$, $\frac{1}{15435536596832413081294738156142102696213782506336175281093721367731} a^{14} - \frac{533476601113871537845113315804721096496465105929452936859953728}{15435536596832413081294738156142102696213782506336175281093721367731} a^{13} - \frac{3941353308656749413215778304986145432756913546620717294098416463}{15435536596832413081294738156142102696213782506336175281093721367731} a^{12} - \frac{81654053979876156390583339062375187629603036765576088033530668728}{15435536596832413081294738156142102696213782506336175281093721367731} a^{11} - \frac{378074007307205684747379785638414030018345693314809150037112144255}{15435536596832413081294738156142102696213782506336175281093721367731} a^{10} - \frac{10639471768285036950573501327370374367741894334555409577668375296}{315010950955763532271321186860042912167628214415023985328443293219} a^{9} + \frac{168227740801508033046274289541832465388442363030717126249260479263}{15435536596832413081294738156142102696213782506336175281093721367731} a^{8} - \frac{348580582847789327006419334197695749962410744918911831167150948375}{15435536596832413081294738156142102696213782506336175281093721367731} a^{7} + \frac{7403768918085168443299109930844570357336113022172553733524042131030}{15435536596832413081294738156142102696213782506336175281093721367731} a^{6} - \frac{951386879488779155802465134987611377267720727377471756908741507053}{15435536596832413081294738156142102696213782506336175281093721367731} a^{5} + \frac{981031744936874564308467591714593059862727471918293201345031459946}{15435536596832413081294738156142102696213782506336175281093721367731} a^{4} + \frac{687085440874496961965498036659929630559315157479791119755067155844}{1403230599712037552844976196012918426928525682394197752826701942521} a^{3} - \frac{189206051956606367871823787953767273160456575026444627638914214846}{2205076656690344725899248308020300385173397500905167897299103052533} a^{2} + \frac{3470666246065518093064671809178200933165317772715431306684888402799}{15435536596832413081294738156142102696213782506336175281093721367731} a - \frac{56147890431314523267487219101055912567449460127505221388837474462}{127566418155639777531361472364810766084411425672199795711518358411}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{30}$, which has order $120$ (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:  \( 817223481214211.9 \) (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

3.1.20300.1, 5.1.5306817753125.4

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 R ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{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$$[5/4]_{12}^{2}$
$7$7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.10.9.2$x^{10} + 14$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$29$29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
29.10.9.1$x^{10} - 29$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$