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Properties

Label 15.15.1832206409...4416.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 3^{20}\cdot 11^{6}\cdot 13^{2}\cdot 157^{6}\cdot 887^{2}\cdot 1907^{2}$
Root discriminant $1415.35$
Ramified primes $2, 3, 11, 13, 157, 887, 1907$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T77

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![332407012923648, 166203506461824, -290856136308192, 56100971904048, 12832994169828, -3229600555881, -178724433528, 61398938061, 707841504, -494319186, 507384, 1758834, -4068, -2421, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2421*x^13 - 4068*x^12 + 1758834*x^11 + 507384*x^10 - 494319186*x^9 + 707841504*x^8 + 61398938061*x^7 - 178724433528*x^6 - 3229600555881*x^5 + 12832994169828*x^4 + 56100971904048*x^3 - 290856136308192*x^2 + 166203506461824*x + 332407012923648)
 
gp: K = bnfinit(x^15 - 2421*x^13 - 4068*x^12 + 1758834*x^11 + 507384*x^10 - 494319186*x^9 + 707841504*x^8 + 61398938061*x^7 - 178724433528*x^6 - 3229600555881*x^5 + 12832994169828*x^4 + 56100971904048*x^3 - 290856136308192*x^2 + 166203506461824*x + 332407012923648, 1)
 

Normalized defining polynomial

\( x^{15} - 2421 x^{13} - 4068 x^{12} + 1758834 x^{11} + 507384 x^{10} - 494319186 x^{9} + 707841504 x^{8} + 61398938061 x^{7} - 178724433528 x^{6} - 3229600555881 x^{5} + 12832994169828 x^{4} + 56100971904048 x^{3} - 290856136308192 x^{2} + 166203506461824 x + 332407012923648 \)

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:  \(183220640975323740990354036775850243373825724416=2^{12}\cdot 3^{20}\cdot 11^{6}\cdot 13^{2}\cdot 157^{6}\cdot 887^{2}\cdot 1907^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1415.35$
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:  $2, 3, 11, 13, 157, 887, 1907$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{18} a^{3} - \frac{1}{2} a$, $\frac{1}{18} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{54} a^{5} - \frac{1}{2} a$, $\frac{1}{324} a^{6} + \frac{1}{12} a^{2}$, $\frac{1}{648} a^{7} - \frac{1}{648} a^{6} - \frac{1}{108} a^{5} - \frac{1}{36} a^{4} + \frac{1}{72} a^{3} + \frac{1}{24} a^{2} - \frac{1}{2} a$, $\frac{1}{1944} a^{8} - \frac{1}{648} a^{6} + \frac{1}{72} a^{4} - \frac{1}{24} a^{2}$, $\frac{1}{11664} a^{9} - \frac{1}{1296} a^{7} - \frac{1}{648} a^{6} - \frac{1}{144} a^{5} - \frac{1}{36} a^{4} + \frac{1}{144} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{11664} a^{10} - \frac{1}{3888} a^{8} - \frac{1}{1296} a^{6} - \frac{1}{144} a^{4} - \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{34992} a^{11} - \frac{1}{648} a^{6} - \frac{1}{108} a^{5} - \frac{1}{36} a^{4} + \frac{1}{144} a^{3} + \frac{1}{24} a^{2}$, $\frac{1}{209952} a^{12} - \frac{1}{3888} a^{8} - \frac{1}{108} a^{5} - \frac{1}{96} a^{4} - \frac{1}{36} a^{3}$, $\frac{1}{209952} a^{13} - \frac{1}{1296} a^{7} - \frac{1}{288} a^{5} - \frac{1}{36} a^{4} - \frac{1}{48} a^{3} + \frac{1}{12} a^{2}$, $\frac{1}{535773009317687838120096646007748392837081539141005778113570624} a^{14} + \frac{11198092105465435968984527892233195564269827873675686321}{11161937694118496627502013458494758184105865398770953710699388} a^{13} + \frac{48318113564516709284069709882089225534038777650986314775}{59530334368631982013344071778638710315231282126778419790396736} a^{12} - \frac{34616693688864547213197388101649240296577101198974686027}{14882583592157995503336017944659677578807820531694604947599184} a^{11} + \frac{69221873046475135669451301806527800680785648769970560845}{3307240798257332334074670654368817239735071229265467766133152} a^{10} - \frac{10590736159521455430405748321264313547378871342035453349}{826810199564333083518667663592204309933767807316366941533288} a^{9} - \frac{761486237811543314663119985997908367308695470399647711593}{3307240798257332334074670654368817239735071229265467766133152} a^{8} + \frac{91329652652755978551433942911791514583916698058915597}{68900849963694423626555638632683692494480650609697245127774} a^{7} - \frac{510752090656135685140003680814687617587916029520637050329}{2204827198838221556049780436245878159823380819510311844088768} a^{6} + \frac{377776925948867702113383711692653106800688812855652657143}{91867799951592564835407518176911589992640867479596326837032} a^{5} - \frac{561609224465268301301129425718669601996582154766405657999}{27220088874545945136417042422788619257078775549510022766528} a^{4} - \frac{570811510907497057998737049959784708975032122850455749887}{61245199967728376556938345451274393328427244986397551224688} a^{3} - \frac{126631510080768590973039533830954636186501684633021505675}{1701255554659121571026065151424288703567423471844376422908} a^{2} + \frac{66611796088533481677221160252548789232272838725531690864}{425313888664780392756516287856072175891855867961094105727} a + \frac{138738732286350072959598794986142323129713035177576034099}{425313888664780392756516287856072175891855867961094105727}$

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:  \( 14281982109800000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T77:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 29160
The 48 conjugacy class representatives for 1/2[3^5:2]S(5)
Character table for 1/2[3^5:2]S(5) is not computed

Intermediate fields

5.5.303952.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ R R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3Data not computed
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.9.6.1$x^{9} - 121 x^{3} + 3993$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$157$157.3.0.1$x^{3} - x + 15$$1$$3$$0$$C_3$$[\ ]^{3}$
157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
157.6.3.1$x^{6} - 314 x^{4} + 24649 x^{2} - 870725925$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
887Data not computed
1907Data not computed